Research ArticleA Predictor-Corrector Method for Solving Equilibrium Problems

Zong-Ke Bao12 Ming Huang23 and Xi-Qiang Xia2

1 School of Accounting Zhejiang University of Finance and Economics Hangzhou 310018 China2Dalian University of Technology Dalian 116024 China3Dongbei University of Finance and Economics Dalian 116025 China

Correspondence should be addressed to Ming Huang huangming0224163com

Received 28 April 2014 Accepted 30 June 2014 Published 15 October 2014

Academic Editor Adrian Petrusel

Copyright copy 2014 Zong-Ke Bao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliaryproblem principle In the main algorithm each stage of computation requires two proximal steps One step serves to predict thenext point the other helps to correct the new prediction At the same time we present convergence analysis under perfect foresightand imperfect one In particular we introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply thisalgorithm for solving the particular case variational inequalities

1 Introduction

Equilibrium problems theory provides us with a unifiednatural innovative and general framework to study a wideclass of problems arising in finance economics networkanalysis transportation elasticity and optimization Thistheory has witnessed an explosive growth in theoreticaladvances and applications across all disciplines of pure andapplied sciences As a result of this interaction we have avariety of techniques to study existence results for equilib-riumproblems see [1ndash4] Equilibriumproblems include vari-ational inequalities as special cases In recent years severalnumerical techniques [5ndash12] including projection resolv-ent and auxiliary principle have been developed and ana-lyzed for solving equilibrium problems

Let 119862 be a nonempty closed convex subset of 119877119899 and let119891 119862times119862 rarr 119877 be a continuous function satisfying 119891(119909 119909) =

0 for all 119909 isin 119862 119891(119909 sdot) is convex on 119862 for all 119909 isin 119862 and119891(sdot 119910) is lower semicontinuous (lsc) on 119862 for all 119910 isin 119862The equilibrium problems (for short EP) proposed by Blum-Oettli [1] are as follows

finding 119909lowast

isin 119862 such that 119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (EP)

Recently much attention has been given to reformulatethe equilibrium problem as an optimization problem Thisproblem is very general in the sense that it includes asparticular cases the optimization problem the variational

inequality problem the Nash equilibrium problem in non-cooperative games the fixed-point problem the nonlinearcomplementarity problem and the vector optimization prob-lem (see eg [1 13] and the references quoted therein)Multiobjective optimization problems can also be obtainedby (EP) as shown by Iusem and Sosa [13] The aboveparticular cases are usefulmodels ofmany practical problemsarising in game theory physics economics and so forthTheinterest of this problem is that it unifies all these particularproblems in a convenientway For example thework of Breziset al extended results concerning variational inequalitiescorresponding to the case where 119891(119909 119910) = ⟨119860119909 119910minus119909⟩ and119860

is a monotone operator (see [14] pages 296-297) Moreovermany methods devoted to solving one of these problems canbe extended with suitable modifications to solve the generalequilibrium problem In this paper we suppose that thereexists at least one solution to problem (EP) In particular itis true when 119862 is compact Other existence results for thisproblem can be found for instance in [1 15]

In this paper one uses usually the auxiliary principle tech-nique This technique deals with finding a suitable auxiliaryproblem and proving that the solution of the auxiliary prob-lem is the solution of the original problem by using thefixed-point approach Glowinski et al [6] used this techniqueto study the existence of a solution of mixed variationalinequalities Noor [8] has used this technique to suggestand analyze a number of iterative methods for solving

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 313217 11 pageshttpdxdoiorg1011552014313217

2 Abstract and Applied Analysis

various classes of variational inequalities It has been shownthat a substantial number of numerical methods can beobtained as special cases from this technique In this paperwe use again the auxiliary principle technique to suggestand analyze some predictor-corrector methods for solvingequilibrium problems In this respect our results representan improvement of previously known results Noor [16] andNoor et al [17] have introduced inertial proximal methodsfor variational inequalities using the auxiliary principle tech-nique and proved that the convergence criteria of inertialproximal methods require only pseudomonotonicity Inertialproximal methods include proximal methods as a specialcase For recent development and applications of the proximalmethods see [5 11 18] Our results can be considered asnovel and important applications of the auxiliary principletechnique This paper is an extension over the related workof [19 20] the main contributions can be summarized asfollows First of all we extend the coefficient of approximatefunction from 120583 isin (0 1] to 120583 isin 119877 0 which is a betterconclusion Secondly approximate function does not need tosatisfy the conditions (1198621)ndash(1198623) in [20] that is to say ourcondition is weaker than thereinMoreover we present a newalgorithm predictor-corrector methods for solving (EP) andgive a stopping criterion In this sense our result representsan improvement and refinement of the known results

We recall the main notations and definitions that will beused in the sequel

A function119891 119862times119862 rarr 119877 is said to be stronglymonotoneon 119862 with modulus 120574 gt 0 if and only if

119891 (119909 119910) + 119891 (119910 119909) le minus1205741003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

forall119909 119910 isin 119862 (1)

A function ℎ 119862 rarr 119877 is said to be strongly convex on 119862

with modulus 120573 (120573 ge 0) if and only if

ℎ (120582119909 + (1 minus 120582) 119910) le120582ℎ (119909) + (1 minus 120582) ℎ (119910)

minus120573

2[120582 (1 minus 120582)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

forall119909 119910 isin 119862 120582 isin [0 1]

(2)

If ℎ is differentiable then ℎ is strongly convex on 119862 withmodulus 120573 (120573 ge 0) if and only if

ℎ (119909) minus ℎ (119910) ge ⟨nablaℎ (119910) 119909 minus 119910⟩ +120573

2

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

forall119909 119910 isin 119862

(3)

A function ℎ 119862 rarr 119877 is said to be Lipschitz continuouson 119862 with modulus 119871 (119871 gt 0) if and only if

1003817100381710038171003817ℎ (119910) minus ℎ (119909)1003817100381710038171003817 le 119871

1003817100381710038171003817119910 minus 1199091003817100381710038171003817 forall119909 119910 isin 119862 (4)

Usually we need there to be at least one solution forequilibrium problems In particular it is true when 119862 iscompact

Proposition 1 (existence of equilibrium (see [19])) Suppose119862 is nonempty compact convex and 119891(119909 119910) is jointly lower

semicontinuous separately continuous in 119909 and convex in 119910Then (EP) admits at least one solution

This paper is organized as follows In Section 2 weintroduce some algorithms In particular we will give apredictor-corrector algorithmic frame We present someconvergence analysis under perfect and imperfect foresightin Section 3 Section 4 is devoted to an application we focuson the particular case variational inequalities problem (VIP)of (EP) mentioned above and we apply our results in theseframeworks and the predictor-corrector algorithm is appliedto (VIP) The paper ends with some concluding remarks

2 Main Algorithm

Most of the algorithms developed for solving EP can bederived from equivalent formulations of the equilibriumproblem We will focus our attention on fixed-point formu-lations of EP we will show that such formulations lead toa generalization of the methods developed by Cohen forvariational inequalities and optimization problems

Let us recall the following preliminary result which statesthe above mentioned equivalent formulation of EP

Lemma 2 Suppose that 119891(119909 119909) = 0 for all 119909 isin 119862 Then thefollowing statements are equivalent

(a) there exists 119909lowast isin 119862 st 119891(119909lowast

119910) ge 0 for all 119910 isin 119862(b) 119909lowast

isin 119862 is a solution of the problem

min119909isin119862

119891 (119909lowast

119909) (5)

We can define the following general iterative algorithmframework

Algorithm 3 Consider the following

Step 1 Set 119896 = 0 1199090 isin 119862

Step 2 Denote by 119909119896+1 the solution of the problem

min119910isin119862

119891(119909119896

119910)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Unfortunately in most of the cases it is not possibleto apply the previous algorithm directly to the equilibriumproblems for the previous algorithm may cause instabilitiesin the iterate process So it is necessary to introduce anauxiliary equilibrium problem which is equivalent to theequilibrium problem

Proposition 4 Let 119891(119909 119910) be a convex differentiable functionwith respect to 119910 at 119909 = 119909

lowast and 120576 gt 0 Let119867(119909 119910) 119862 times 119862 rarr

119877 be a nonnegative differentiable function on the convex set 119862with respect to 119910 and such that

(i) 119867(119909 119909) = 0 for all 119909 isin 119862(ii) 119867

1015840

119910(119909 119909) = 0 for all 119909 isin 119862

Abstract and Applied Analysis 3

Then 119909lowast is a solution of EP if and only if it is a solution of the

auxiliary equilibrium problem (AEP)

119891119894119899119889119894119899119892 119909lowast

isin 119862 119904119905 120576119891 (119909lowast

119910) + 119867 (119909lowast

119910) ge 0 forall119910 isin 119862

(AEP)

Proof It is easy to know that if 119909lowast is a solution of EP then itis also a solution of AEP

Vice versa let 119909lowast be a solution of AEP Then 119909

lowast is aminimum point of the problem

min119909isin119870

[120576119891 (119909lowast

119910) + 119867 (119909lowast

119910)] (6)

Because 119870 is convex then 119909lowast is an optimal solution for (6) if

and only if

⟨1205761198911015840

119910(119909lowast

119909lowast

) + 1198671015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (7)

so that

⟨1205761198911015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (8)

Dividing by 120576 we obtain that (8) implies by the convexity of119891(119909lowast

sdot) that

119891 (119909lowast

119910) ge 119891 (119909lowast

119909lowast

) = 0 forall119910 isin 119870 (9)

Remark 5 Suppose ℎ 119862 rarr 119877 is a strongly convexdifferentiable function denote 119867(119909 119910) = ℎ(119910) minus ℎ(119909) minus

⟨nablaℎ(119909) 119910 minus 119909⟩ for all 119909 119910 isin 119862 We have

119867(119909 119910) = ℎ (119910) minus ℎ (119909) minus ⟨nablaℎ (119909) 119910 minus 119909⟩

ge120573

2

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

ge 0

119867 (119909 119909) = 0

1198671015840

119910(119909 119909) = 0

(10)

That is119867(119909 119910) satisfies Proposition 4

Applying Algorithm 3 to the AEP we obtain the followingiterative method

Algorithm 6 Consider the following

Step 1 Set 119896 = 0 1199090 isin 119862

Step 2 Denote by 119909119896+1 the solution of the problem

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Most papers about EP only study the existence of EPrsquossolution In this paper we will give a predictor-correctormethod to solve the equilibrium problems

Definition 7 Let 120583 isin 119877 0 and 119909 isin 119862 A convex function119891(119909 sdot) 119862 rarr 119877 is a 120583-approximation of 119891(119909 sdot) at 119909 if119891(119909 sdot) le 119891(119909 sdot) on 119862 and 119891(119909 119910) le 120583119891(119909 119910) where 119910 =

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Remark 8 According to the above we extend the coefficientof approximate function from 120583 isin (0 1] in [20] to 120583 isin 119877 0which is a more generic case

Now we describe the framework of predictor-correctoralgorithm as follows

Algorithm 9 Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862

Step 2 Find 120583-approximation of 119891(119909 sdot) at 119909 119891(119909 sdot) bypredictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(11)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Remark 10 In Algorithm 9 each stage of computationrequires two proximal steps In Step 2 119909

119896+ is served topredict the next point the other 119909119896+1 helps to correct the newprediction

3 Convergence Analysis

In this section we will give some convergence results aboutthe algorithm

Definition 11 In Algorithm 9 if 119909119896+1 = 119909119896+ 119909119896+ is called a

perfect foresight point of 119909119896+1 otherwise 119909119896+ is an imperfect

foresight point of 119909119896+1

Next we give the convergence result under perfect fore-sight which has been stated in [20]

Proposition 12 (see [20]) Assume that there exist numbers119903 119888 119889 gt 0 and a nonnegative function 119892 119862 times 119862 rarr 119877 suchthat for all 119909 119910 119911 isin 119862

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119892(119910 119909)(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and 120572119896le 1205731205832119889 for

all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 13 (see [20]) Assume that 120572119896ge 120572 gt 0 for all 119896 isin

119873 If the sequence 119909119896119896isin119873

generated by the predictor-corrector

4 Abstract and Applied Analysis

algorithm is bounded and lim119896rarrinfin

119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of problem (EP)

At the same time respective to convergence under imper-fect foresight we first give some denotations and results

By the previous introduction we have

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910)

minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

(12)

119909119896+1

isin 120576119896minus argmin

119910isin119862

120572119896119891 (119909119896+

119910)

+ ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910⟩

(13)

Using (12) and (13) we get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

le 120572119896119891 (119909

119896

119910) minus 119891 (119909119896

119909119896+

) forall119910 isin 119862

(14)

⟨nablaℎ (119909119896+

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896 forall119910 isin 119862

(15)

Arranging (15) we have

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+1

⟩

(16)

Let 119910 = 119909lowast in (14) and (16) then adding them we can get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus 119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

+ 119891 (119909119896

119909lowast

) minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(17)

(119891 (119909119896

sdot) le 119891 (119909119896

sdot)

on 119862 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) at 119909119896+

)

(18)

Assumption 14 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 + (4119888120583) + (2119871120572) (5119888120583) + (2119871120572) for all 119909 119910 119911 isin 119862Consider the following

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119909 minus 1199102

(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119909 minus 1199102

+ 119889119911 minus 1199102

We denote

119860119896= 2120583120572

119896119888 + 2120572

119896119889 + 120583119871

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 3120583119871 + 4120572

119896119888

119862119896=

2120572119896119889

120583

119864119896= min

119903 minus 119889

4minus

119888

120583minus

119871

2120572119896

120573 minus 119861119896

4120583120572119896

120572119896119903120583 minus 5120572

119896119888 minus 2120583119871

4120583120572119896

(19)

It is convenient to prove the following theorem

Theorem 15 Assume that the function 119891(119909 119910) satisfiesAssumption 14 and120573 ge max 119860

119896 119861119896 119862119896 120576119896le 119864119896119909119896

minus 119909119896+

2

then the sequence 119909119896

119896isin119873generated by the predictor-corrector

methods is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proof Let 119909lowast be a solution of (EP) and consider for each 119896 isin

119873 the Lyapunov function Γ119896

119862 times 119862 rarr 119877 defined for all119910 119911 isin 119862

Γ119896

(119910 119911) = ℎ (119911) minus ℎ (119910) minus ⟨nablaℎ (119910) 119911 minus 119910⟩ +120572119896

120583119891 (119911 119910)

(20)

Since ℎ is strongly convex on 119862 with modulus 120573 we caneasily obtain that for all 119909119896 isin 119862

Γ119896

(119909119896

119909lowast

) ge120573

2

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

(21)

Consider the following relation

Γ119896+1

(119909119896+1

119909lowast

) minus Γ119896

(119909119896

119909lowast

)

le ℎ (119909119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

= 1198781+ 1198782+ 1198783

(22)

where

1198781= ℎ (119909

119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

1198782= ⟨nablaℎ (119909

119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

1198783=

120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

(23)

Abstract and Applied Analysis 5

For 1198781 we can easily get the following from the strong

convexity of ℎ

1198781le minus

120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(24)

For 1198782 we derive the following from (17)

1198782le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(25)

Then

1198782+ 1198783le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

= 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

) + 120576119896

+120572119896

120583119891 (119909

119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(26)

For the last term on the right of the above equality wehave

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(27)

We can obtain the following from assumption (ii)

119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

= 119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) minus 119891 (119909119896+1

119909lowast

) + 119891 (119909119896+1

119909lowast

)

le 11988810038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

(28)

Similarly

119891 (119909lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

le 11988810038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

= 119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896+

119909119896+1

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

(29)

Because of 120572119896119891(119909119896+

119909119896+1

) le minus(1205732)119909119896+1

minus 119909119896+

2

wederive the following from (13)

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le 120572119896119891 (119909119896+

119910) + ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

forall119910 isin 119862

(30)

In particular let 119910 = 119909119896+ we have

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 119891 (119909119896+

119909119896+

) = 0

(31)

That is

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 0

lArrrArr 120572119896119891 (119909119896+

119909119896+1

) le minusℎ (119909119896+1

) + ℎ (119909119896+

)

+ ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(32)

Hence

120572119896119891 (119909119896+

119909119896+1

) le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(33)

Finally we obtain

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(34)

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

various classes of variational inequalities It has been shownthat a substantial number of numerical methods can beobtained as special cases from this technique In this paperwe use again the auxiliary principle technique to suggestand analyze some predictor-corrector methods for solvingequilibrium problems In this respect our results representan improvement of previously known results Noor [16] andNoor et al [17] have introduced inertial proximal methodsfor variational inequalities using the auxiliary principle tech-nique and proved that the convergence criteria of inertialproximal methods require only pseudomonotonicity Inertialproximal methods include proximal methods as a specialcase For recent development and applications of the proximalmethods see [5 11 18] Our results can be considered asnovel and important applications of the auxiliary principletechnique This paper is an extension over the related workof [19 20] the main contributions can be summarized asfollows First of all we extend the coefficient of approximatefunction from 120583 isin (0 1] to 120583 isin 119877 0 which is a betterconclusion Secondly approximate function does not need tosatisfy the conditions (1198621)ndash(1198623) in [20] that is to say ourcondition is weaker than thereinMoreover we present a newalgorithm predictor-corrector methods for solving (EP) andgive a stopping criterion In this sense our result representsan improvement and refinement of the known results

We recall the main notations and definitions that will beused in the sequel

A function119891 119862times119862 rarr 119877 is said to be stronglymonotoneon 119862 with modulus 120574 gt 0 if and only if

119891 (119909 119910) + 119891 (119910 119909) le minus1205741003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

forall119909 119910 isin 119862 (1)

A function ℎ 119862 rarr 119877 is said to be strongly convex on 119862

with modulus 120573 (120573 ge 0) if and only if

ℎ (120582119909 + (1 minus 120582) 119910) le120582ℎ (119909) + (1 minus 120582) ℎ (119910)

minus120573

2[120582 (1 minus 120582)]

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

forall119909 119910 isin 119862 120582 isin [0 1]

(2)

If ℎ is differentiable then ℎ is strongly convex on 119862 withmodulus 120573 (120573 ge 0) if and only if

ℎ (119909) minus ℎ (119910) ge ⟨nablaℎ (119910) 119909 minus 119910⟩ +120573

2

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

forall119909 119910 isin 119862

(3)

A function ℎ 119862 rarr 119877 is said to be Lipschitz continuouson 119862 with modulus 119871 (119871 gt 0) if and only if

1003817100381710038171003817ℎ (119910) minus ℎ (119909)1003817100381710038171003817 le 119871

1003817100381710038171003817119910 minus 1199091003817100381710038171003817 forall119909 119910 isin 119862 (4)

Usually we need there to be at least one solution forequilibrium problems In particular it is true when 119862 iscompact

Proposition 1 (existence of equilibrium (see [19])) Suppose119862 is nonempty compact convex and 119891(119909 119910) is jointly lower

semicontinuous separately continuous in 119909 and convex in 119910Then (EP) admits at least one solution

This paper is organized as follows In Section 2 weintroduce some algorithms In particular we will give apredictor-corrector algorithmic frame We present someconvergence analysis under perfect and imperfect foresightin Section 3 Section 4 is devoted to an application we focuson the particular case variational inequalities problem (VIP)of (EP) mentioned above and we apply our results in theseframeworks and the predictor-corrector algorithm is appliedto (VIP) The paper ends with some concluding remarks

2 Main Algorithm

Most of the algorithms developed for solving EP can bederived from equivalent formulations of the equilibriumproblem We will focus our attention on fixed-point formu-lations of EP we will show that such formulations lead toa generalization of the methods developed by Cohen forvariational inequalities and optimization problems

Let us recall the following preliminary result which statesthe above mentioned equivalent formulation of EP

Lemma 2 Suppose that 119891(119909 119909) = 0 for all 119909 isin 119862 Then thefollowing statements are equivalent

(a) there exists 119909lowast isin 119862 st 119891(119909lowast

119910) ge 0 for all 119910 isin 119862(b) 119909lowast

isin 119862 is a solution of the problem

min119909isin119862

119891 (119909lowast

119909) (5)

We can define the following general iterative algorithmframework

Algorithm 3 Consider the following

Step 1 Set 119896 = 0 1199090 isin 119862

Step 2 Denote by 119909119896+1 the solution of the problem

min119910isin119862

119891(119909119896

119910)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Unfortunately in most of the cases it is not possibleto apply the previous algorithm directly to the equilibriumproblems for the previous algorithm may cause instabilitiesin the iterate process So it is necessary to introduce anauxiliary equilibrium problem which is equivalent to theequilibrium problem

Proposition 4 Let 119891(119909 119910) be a convex differentiable functionwith respect to 119910 at 119909 = 119909

lowast and 120576 gt 0 Let119867(119909 119910) 119862 times 119862 rarr

119877 be a nonnegative differentiable function on the convex set 119862with respect to 119910 and such that

(i) 119867(119909 119909) = 0 for all 119909 isin 119862(ii) 119867

1015840

119910(119909 119909) = 0 for all 119909 isin 119862

Abstract and Applied Analysis 3

Then 119909lowast is a solution of EP if and only if it is a solution of the

auxiliary equilibrium problem (AEP)

119891119894119899119889119894119899119892 119909lowast

isin 119862 119904119905 120576119891 (119909lowast

119910) + 119867 (119909lowast

119910) ge 0 forall119910 isin 119862

(AEP)

Proof It is easy to know that if 119909lowast is a solution of EP then itis also a solution of AEP

Vice versa let 119909lowast be a solution of AEP Then 119909

lowast is aminimum point of the problem

min119909isin119870

[120576119891 (119909lowast

119910) + 119867 (119909lowast

119910)] (6)

Because 119870 is convex then 119909lowast is an optimal solution for (6) if

and only if

⟨1205761198911015840

119910(119909lowast

119909lowast

) + 1198671015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (7)

so that

⟨1205761198911015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (8)

Dividing by 120576 we obtain that (8) implies by the convexity of119891(119909lowast

sdot) that

119891 (119909lowast

119910) ge 119891 (119909lowast

119909lowast

) = 0 forall119910 isin 119870 (9)

Remark 5 Suppose ℎ 119862 rarr 119877 is a strongly convexdifferentiable function denote 119867(119909 119910) = ℎ(119910) minus ℎ(119909) minus

⟨nablaℎ(119909) 119910 minus 119909⟩ for all 119909 119910 isin 119862 We have

119867(119909 119910) = ℎ (119910) minus ℎ (119909) minus ⟨nablaℎ (119909) 119910 minus 119909⟩

ge120573

2

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

ge 0

119867 (119909 119909) = 0

1198671015840

119910(119909 119909) = 0

(10)

That is119867(119909 119910) satisfies Proposition 4

Applying Algorithm 3 to the AEP we obtain the followingiterative method

Algorithm 6 Consider the following

Step 1 Set 119896 = 0 1199090 isin 119862

Step 2 Denote by 119909119896+1 the solution of the problem

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Most papers about EP only study the existence of EPrsquossolution In this paper we will give a predictor-correctormethod to solve the equilibrium problems

Definition 7 Let 120583 isin 119877 0 and 119909 isin 119862 A convex function119891(119909 sdot) 119862 rarr 119877 is a 120583-approximation of 119891(119909 sdot) at 119909 if119891(119909 sdot) le 119891(119909 sdot) on 119862 and 119891(119909 119910) le 120583119891(119909 119910) where 119910 =

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Remark 8 According to the above we extend the coefficientof approximate function from 120583 isin (0 1] in [20] to 120583 isin 119877 0which is a more generic case

Now we describe the framework of predictor-correctoralgorithm as follows

Algorithm 9 Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862

Step 2 Find 120583-approximation of 119891(119909 sdot) at 119909 119891(119909 sdot) bypredictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(11)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Remark 10 In Algorithm 9 each stage of computationrequires two proximal steps In Step 2 119909

119896+ is served topredict the next point the other 119909119896+1 helps to correct the newprediction

3 Convergence Analysis

In this section we will give some convergence results aboutthe algorithm

Definition 11 In Algorithm 9 if 119909119896+1 = 119909119896+ 119909119896+ is called a

perfect foresight point of 119909119896+1 otherwise 119909119896+ is an imperfect

foresight point of 119909119896+1

Next we give the convergence result under perfect fore-sight which has been stated in [20]

Proposition 12 (see [20]) Assume that there exist numbers119903 119888 119889 gt 0 and a nonnegative function 119892 119862 times 119862 rarr 119877 suchthat for all 119909 119910 119911 isin 119862

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119892(119910 119909)(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and 120572119896le 1205731205832119889 for

all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 13 (see [20]) Assume that 120572119896ge 120572 gt 0 for all 119896 isin

119873 If the sequence 119909119896119896isin119873

generated by the predictor-corrector

4 Abstract and Applied Analysis

algorithm is bounded and lim119896rarrinfin

119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of problem (EP)

At the same time respective to convergence under imper-fect foresight we first give some denotations and results

By the previous introduction we have

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910)

minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

(12)

119909119896+1

isin 120576119896minus argmin

119910isin119862

120572119896119891 (119909119896+

119910)

+ ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910⟩

(13)

Using (12) and (13) we get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

le 120572119896119891 (119909

119896

119910) minus 119891 (119909119896

119909119896+

) forall119910 isin 119862

(14)

⟨nablaℎ (119909119896+

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896 forall119910 isin 119862

(15)

Arranging (15) we have

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+1

⟩

(16)

Let 119910 = 119909lowast in (14) and (16) then adding them we can get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus 119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

+ 119891 (119909119896

119909lowast

) minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(17)

(119891 (119909119896

sdot) le 119891 (119909119896

sdot)

on 119862 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) at 119909119896+

)

(18)

Assumption 14 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 + (4119888120583) + (2119871120572) (5119888120583) + (2119871120572) for all 119909 119910 119911 isin 119862Consider the following

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119909 minus 1199102

(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119909 minus 1199102

+ 119889119911 minus 1199102

We denote

119860119896= 2120583120572

119896119888 + 2120572

119896119889 + 120583119871

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 3120583119871 + 4120572

119896119888

119862119896=

2120572119896119889

120583

119864119896= min

119903 minus 119889

4minus

119888

120583minus

119871

2120572119896

120573 minus 119861119896

4120583120572119896

120572119896119903120583 minus 5120572

119896119888 minus 2120583119871

4120583120572119896

(19)

It is convenient to prove the following theorem

Theorem 15 Assume that the function 119891(119909 119910) satisfiesAssumption 14 and120573 ge max 119860

119896 119861119896 119862119896 120576119896le 119864119896119909119896

minus 119909119896+

2

then the sequence 119909119896

119896isin119873generated by the predictor-corrector

methods is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proof Let 119909lowast be a solution of (EP) and consider for each 119896 isin

119873 the Lyapunov function Γ119896

119862 times 119862 rarr 119877 defined for all119910 119911 isin 119862

Γ119896

(119910 119911) = ℎ (119911) minus ℎ (119910) minus ⟨nablaℎ (119910) 119911 minus 119910⟩ +120572119896

120583119891 (119911 119910)

(20)

Since ℎ is strongly convex on 119862 with modulus 120573 we caneasily obtain that for all 119909119896 isin 119862

Γ119896

(119909119896

119909lowast

) ge120573

2

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

(21)

Consider the following relation

Γ119896+1

(119909119896+1

119909lowast

) minus Γ119896

(119909119896

119909lowast

)

le ℎ (119909119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

= 1198781+ 1198782+ 1198783

(22)

where

1198781= ℎ (119909

119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

1198782= ⟨nablaℎ (119909

119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

1198783=

120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

(23)

Abstract and Applied Analysis 5

For 1198781 we can easily get the following from the strong

convexity of ℎ

1198781le minus

120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(24)

For 1198782 we derive the following from (17)

1198782le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(25)

Then

1198782+ 1198783le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

= 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

) + 120576119896

+120572119896

120583119891 (119909

119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(26)

For the last term on the right of the above equality wehave

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(27)

We can obtain the following from assumption (ii)

119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

= 119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) minus 119891 (119909119896+1

119909lowast

) + 119891 (119909119896+1

119909lowast

)

le 11988810038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

(28)

Similarly

119891 (119909lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

le 11988810038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

= 119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896+

119909119896+1

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

(29)

Because of 120572119896119891(119909119896+

119909119896+1

) le minus(1205732)119909119896+1

minus 119909119896+

2

wederive the following from (13)

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le 120572119896119891 (119909119896+

119910) + ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

forall119910 isin 119862

(30)

In particular let 119910 = 119909119896+ we have

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 119891 (119909119896+

119909119896+

) = 0

(31)

That is

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 0

lArrrArr 120572119896119891 (119909119896+

119909119896+1

) le minusℎ (119909119896+1

) + ℎ (119909119896+

)

+ ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(32)

Hence

120572119896119891 (119909119896+

119909119896+1

) le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(33)

Finally we obtain

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(34)

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Then 119909lowast is a solution of EP if and only if it is a solution of the

auxiliary equilibrium problem (AEP)

119891119894119899119889119894119899119892 119909lowast

isin 119862 119904119905 120576119891 (119909lowast

119910) + 119867 (119909lowast

119910) ge 0 forall119910 isin 119862

(AEP)

Proof It is easy to know that if 119909lowast is a solution of EP then itis also a solution of AEP

Vice versa let 119909lowast be a solution of AEP Then 119909

lowast is aminimum point of the problem

min119909isin119870

[120576119891 (119909lowast

119910) + 119867 (119909lowast

119910)] (6)

Because 119870 is convex then 119909lowast is an optimal solution for (6) if

and only if

⟨1205761198911015840

119910(119909lowast

119909lowast

) + 1198671015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (7)

so that

⟨1205761198911015840

119910(119909lowast

119909lowast

) 119909lowast

minus 119910⟩ ge 0 forall119910 isin 119870 (8)

Dividing by 120576 we obtain that (8) implies by the convexity of119891(119909lowast

sdot) that

119891 (119909lowast

119910) ge 119891 (119909lowast

119909lowast

) = 0 forall119910 isin 119870 (9)

Remark 5 Suppose ℎ 119862 rarr 119877 is a strongly convexdifferentiable function denote 119867(119909 119910) = ℎ(119910) minus ℎ(119909) minus

⟨nablaℎ(119909) 119910 minus 119909⟩ for all 119909 119910 isin 119862 We have

119867(119909 119910) = ℎ (119910) minus ℎ (119909) minus ⟨nablaℎ (119909) 119910 minus 119909⟩

ge120573

2

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

ge 0

119867 (119909 119909) = 0

1198671015840

119910(119909 119909) = 0

(10)

That is119867(119909 119910) satisfies Proposition 4

Applying Algorithm 3 to the AEP we obtain the followingiterative method

Algorithm 6 Consider the following

Step 1 Set 119896 = 0 1199090 isin 119862

Step 2 Denote by 119909119896+1 the solution of the problem

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Most papers about EP only study the existence of EPrsquossolution In this paper we will give a predictor-correctormethod to solve the equilibrium problems

Definition 7 Let 120583 isin 119877 0 and 119909 isin 119862 A convex function119891(119909 sdot) 119862 rarr 119877 is a 120583-approximation of 119891(119909 sdot) at 119909 if119891(119909 sdot) le 119891(119909 sdot) on 119862 and 119891(119909 119910) le 120583119891(119909 119910) where 119910 =

min119910isin119862

120576119891(119909119896

119910) + 119867(119909119896

119910)

Remark 8 According to the above we extend the coefficientof approximate function from 120583 isin (0 1] in [20] to 120583 isin 119877 0which is a more generic case

Now we describe the framework of predictor-correctoralgorithm as follows

Algorithm 9 Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862

Step 2 Find 120583-approximation of 119891(119909 sdot) at 119909 119891(119909 sdot) bypredictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(11)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 for some fixed 120575 gt 0 then stopotherwise let 119896 = 119896 + 1 and go to Step 2

Remark 10 In Algorithm 9 each stage of computationrequires two proximal steps In Step 2 119909

119896+ is served topredict the next point the other 119909119896+1 helps to correct the newprediction

3 Convergence Analysis

In this section we will give some convergence results aboutthe algorithm

Definition 11 In Algorithm 9 if 119909119896+1 = 119909119896+ 119909119896+ is called a

perfect foresight point of 119909119896+1 otherwise 119909119896+ is an imperfect

foresight point of 119909119896+1

Next we give the convergence result under perfect fore-sight which has been stated in [20]

Proposition 12 (see [20]) Assume that there exist numbers119903 119888 119889 gt 0 and a nonnegative function 119892 119862 times 119862 rarr 119877 suchthat for all 119909 119910 119911 isin 119862

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119892(119910 119909)(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and 120572119896le 1205731205832119889 for

all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 13 (see [20]) Assume that 120572119896ge 120572 gt 0 for all 119896 isin

119873 If the sequence 119909119896119896isin119873

generated by the predictor-corrector

4 Abstract and Applied Analysis

algorithm is bounded and lim119896rarrinfin

119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of problem (EP)

At the same time respective to convergence under imper-fect foresight we first give some denotations and results

By the previous introduction we have

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910)

minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

(12)

119909119896+1

isin 120576119896minus argmin

119910isin119862

120572119896119891 (119909119896+

119910)

+ ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910⟩

(13)

Using (12) and (13) we get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

le 120572119896119891 (119909

119896

119910) minus 119891 (119909119896

119909119896+

) forall119910 isin 119862

(14)

⟨nablaℎ (119909119896+

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896 forall119910 isin 119862

(15)

Arranging (15) we have

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+1

⟩

(16)

Let 119910 = 119909lowast in (14) and (16) then adding them we can get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus 119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

+ 119891 (119909119896

119909lowast

) minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(17)

(119891 (119909119896

sdot) le 119891 (119909119896

sdot)

on 119862 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) at 119909119896+

)

(18)

Assumption 14 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 + (4119888120583) + (2119871120572) (5119888120583) + (2119871120572) for all 119909 119910 119911 isin 119862Consider the following

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119909 minus 1199102

(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119909 minus 1199102

+ 119889119911 minus 1199102

We denote

119860119896= 2120583120572

119896119888 + 2120572

119896119889 + 120583119871

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 3120583119871 + 4120572

119896119888

119862119896=

2120572119896119889

120583

119864119896= min

119903 minus 119889

4minus

119888

120583minus

119871

2120572119896

120573 minus 119861119896

4120583120572119896

120572119896119903120583 minus 5120572

119896119888 minus 2120583119871

4120583120572119896

(19)

It is convenient to prove the following theorem

Theorem 15 Assume that the function 119891(119909 119910) satisfiesAssumption 14 and120573 ge max 119860

119896 119861119896 119862119896 120576119896le 119864119896119909119896

minus 119909119896+

2

then the sequence 119909119896

119896isin119873generated by the predictor-corrector

methods is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proof Let 119909lowast be a solution of (EP) and consider for each 119896 isin

119873 the Lyapunov function Γ119896

119862 times 119862 rarr 119877 defined for all119910 119911 isin 119862

Γ119896

(119910 119911) = ℎ (119911) minus ℎ (119910) minus ⟨nablaℎ (119910) 119911 minus 119910⟩ +120572119896

120583119891 (119911 119910)

(20)

Since ℎ is strongly convex on 119862 with modulus 120573 we caneasily obtain that for all 119909119896 isin 119862

Γ119896

(119909119896

119909lowast

) ge120573

2

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

(21)

Consider the following relation

Γ119896+1

(119909119896+1

119909lowast

) minus Γ119896

(119909119896

119909lowast

)

le ℎ (119909119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

= 1198781+ 1198782+ 1198783

(22)

where

1198781= ℎ (119909

119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

1198782= ⟨nablaℎ (119909

119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

1198783=

120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

(23)

Abstract and Applied Analysis 5

For 1198781 we can easily get the following from the strong

convexity of ℎ

1198781le minus

120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(24)

For 1198782 we derive the following from (17)

1198782le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(25)

Then

1198782+ 1198783le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

= 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

) + 120576119896

+120572119896

120583119891 (119909

119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(26)

For the last term on the right of the above equality wehave

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(27)

We can obtain the following from assumption (ii)

119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

= 119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) minus 119891 (119909119896+1

119909lowast

) + 119891 (119909119896+1

119909lowast

)

le 11988810038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

(28)

Similarly

119891 (119909lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

le 11988810038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

= 119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896+

119909119896+1

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

(29)

Because of 120572119896119891(119909119896+

119909119896+1

) le minus(1205732)119909119896+1

minus 119909119896+

2

wederive the following from (13)

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le 120572119896119891 (119909119896+

119910) + ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

forall119910 isin 119862

(30)

In particular let 119910 = 119909119896+ we have

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 119891 (119909119896+

119909119896+

) = 0

(31)

That is

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 0

lArrrArr 120572119896119891 (119909119896+

119909119896+1

) le minusℎ (119909119896+1

) + ℎ (119909119896+

)

+ ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(32)

Hence

120572119896119891 (119909119896+

119909119896+1

) le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(33)

Finally we obtain

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(34)

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

algorithm is bounded and lim119896rarrinfin

119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of problem (EP)

At the same time respective to convergence under imper-fect foresight we first give some denotations and results

By the previous introduction we have

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910)

minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

(12)

119909119896+1

isin 120576119896minus argmin

119910isin119862

120572119896119891 (119909119896+

119910)

+ ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910⟩

(13)

Using (12) and (13) we get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

le 120572119896119891 (119909

119896

119910) minus 119891 (119909119896

119909119896+

) forall119910 isin 119862

(14)

⟨nablaℎ (119909119896+

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896 forall119910 isin 119862

(15)

Arranging (15) we have

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119910 minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119910) minus 119891 (119909119896+

119909119896+1

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+1

⟩

(16)

Let 119910 = 119909lowast in (14) and (16) then adding them we can get

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus 119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

+ 119891 (119909119896

119909lowast

) minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(17)

(119891 (119909119896

sdot) le 119891 (119909119896

sdot)

on 119862 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) at 119909119896+

)

(18)

Assumption 14 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 + (4119888120583) + (2119871120572) (5119888120583) + (2119871120572) for all 119909 119910 119911 isin 119862Consider the following

(i) 119891(119909 119910) ge 0 rArr 119891(119910 119909) le minus119903119909 minus 1199102

(ii) 119891(119909 119911) minus 119891(119910 119911) minus 119891(119909 119910) le 119888119909 minus 1199102

+ 119889119911 minus 1199102

We denote

119860119896= 2120583120572

119896119888 + 2120572

119896119889 + 120583119871

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 3120583119871 + 4120572

119896119888

119862119896=

2120572119896119889

120583

119864119896= min

119903 minus 119889

4minus

119888

120583minus

119871

2120572119896

120573 minus 119861119896

4120583120572119896

120572119896119903120583 minus 5120572

119896119888 minus 2120583119871

4120583120572119896

(19)

It is convenient to prove the following theorem

Theorem 15 Assume that the function 119891(119909 119910) satisfiesAssumption 14 and120573 ge max 119860

119896 119861119896 119862119896 120576119896le 119864119896119909119896

minus 119909119896+

2

then the sequence 119909119896

119896isin119873generated by the predictor-corrector

methods is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proof Let 119909lowast be a solution of (EP) and consider for each 119896 isin

119873 the Lyapunov function Γ119896

119862 times 119862 rarr 119877 defined for all119910 119911 isin 119862

Γ119896

(119910 119911) = ℎ (119911) minus ℎ (119910) minus ⟨nablaℎ (119910) 119911 minus 119910⟩ +120572119896

120583119891 (119911 119910)

(20)

Since ℎ is strongly convex on 119862 with modulus 120573 we caneasily obtain that for all 119909119896 isin 119862

Γ119896

(119909119896

119909lowast

) ge120573

2

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

(21)

Consider the following relation

Γ119896+1

(119909119896+1

119909lowast

) minus Γ119896

(119909119896

119909lowast

)

le ℎ (119909119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

= 1198781+ 1198782+ 1198783

(22)

where

1198781= ℎ (119909

119896

) minus ℎ (119909119896+1

) + ⟨nablaℎ (119909119896

) 119909119896+1

minus 119909119896

⟩

1198782= ⟨nablaℎ (119909

119896

) minus nablaℎ (119909119896+1

) 119909lowast

minus 119909119896+1

⟩

1198783=

120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

(23)

Abstract and Applied Analysis 5

For 1198781 we can easily get the following from the strong

convexity of ℎ

1198781le minus

120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(24)

For 1198782 we derive the following from (17)

1198782le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(25)

Then

1198782+ 1198783le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

= 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

) + 120576119896

+120572119896

120583119891 (119909

119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(26)

For the last term on the right of the above equality wehave

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(27)

We can obtain the following from assumption (ii)

119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

= 119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) minus 119891 (119909119896+1

119909lowast

) + 119891 (119909119896+1

119909lowast

)

le 11988810038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

(28)

Similarly

119891 (119909lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

le 11988810038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

= 119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896+

119909119896+1

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

(29)

Because of 120572119896119891(119909119896+

119909119896+1

) le minus(1205732)119909119896+1

minus 119909119896+

2

wederive the following from (13)

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le 120572119896119891 (119909119896+

119910) + ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

forall119910 isin 119862

(30)

In particular let 119910 = 119909119896+ we have

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 119891 (119909119896+

119909119896+

) = 0

(31)

That is

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 0

lArrrArr 120572119896119891 (119909119896+

119909119896+1

) le minusℎ (119909119896+1

) + ℎ (119909119896+

)

+ ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(32)

Hence

120572119896119891 (119909119896+

119909119896+1

) le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(33)

Finally we obtain

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(34)

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

For 1198781 we can easily get the following from the strong

convexity of ℎ

1198781le minus

120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(24)

For 1198782 we derive the following from (17)

1198782le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(25)

Then

1198782+ 1198783le 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

)

minus1

120583119891 (119909119896

119909119896+

) + 120576119896

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

= 120572119896119891 (119909

119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896

119909lowast

) + 120576119896

+120572119896

120583119891 (119909

119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

+120572119896

120583119891 (119909

lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

+ ⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

(26)

For the last term on the right of the above equality wehave

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119909119896+

minus 119909119896+1

⟩

le119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(27)

We can obtain the following from assumption (ii)

119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

)

= 119891 (119909119896+

119909lowast

) minus 119891 (119909119896+

119909119896+1

) minus 119891 (119909119896+1

119909lowast

) + 119891 (119909119896+1

119909lowast

)

le 11988810038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

(28)

Similarly

119891 (119909lowast

119909119896+1

) minus 119891 (119909lowast

119909119896

) minus 119891 (119909119896

119909119896+1

)

le 11988810038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

= 119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

minus 119891 (119909119896+

119909119896+1

) + 119891 (119909119896+

119909119896+1

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

(29)

Because of 120572119896119891(119909119896+

119909119896+1

) le minus(1205732)119909119896+1

minus 119909119896+

2

wederive the following from (13)

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le 120572119896119891 (119909119896+

119910) + ℎ (119910) minus ℎ (119909119896+

) minus ⟨nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩

forall119910 isin 119862

(30)

In particular let 119910 = 119909119896+ we have

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 119891 (119909119896+

119909119896+

) = 0

(31)

That is

120572119896119891 (119909119896+

119909119896+1

) + ℎ (119909119896+1

) minus ℎ (119909119896+

)

minus ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩ le 0

lArrrArr 120572119896119891 (119909119896+

119909119896+1

) le minusℎ (119909119896+1

) + ℎ (119909119896+

)

+ ⟨nablaℎ (119909119896+

) 119909119896+1

minus 119909119896+

⟩

le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(32)

Hence

120572119896119891 (119909119896+

119909119896+1

) le minus120573

2

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(33)

Finally we obtain

119891 (119909119896

119909119896+1

) minus 119891 (119909119896

119909119896+

)

le 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

(34)

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

So

1198781+ 1198782+ 1198783

le 120572119896119888

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus 119903)10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

minus 11990310038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ 120576119896

+120572119896

120583119888

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+120572119896

120583119888

10038171003817100381710038171003817119909lowast

minus 11990911989610038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+119871

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

minus120573

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(35)

Arrange the previous inequality relation we can get

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ (120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)

times10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

times10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(36)

Under the condition of (1205732) minus (120572119896119889120583) gt 0 in order to

obtain 1198781+ 1198782+ 1198783le 0 we only need to prove the following

result

(120572119896119888 +

120572119896119889

120583+

119871

2minus

120573

2120583)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896(119889 minus 119903)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119888

120583minus 119903)

10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120572119896119888

120583+

119871

2)

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896le 0

(37)

That is

(120572119896119888

120583+

119871

2)10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

le 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(38)

(1)When 120583119889 ge 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(39)

Then

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

+ (120573

2120583minus 120572119896119888 minus

120572119896119889

120583minus

119871

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(40)

We discuss in two cases(1∘

) If (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119889)2

120572119896(119903 minus 119889)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+120573 minus 119860

119896

2120583

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

210038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

ge120572119896(119903 minus 119889)

4

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

2

(41)

So

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(42)

We know that

120573

2minus

120572119896119889

120583gt 0

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2gt 0

120572119896120576119896le [

120572119896(119903 minus 119889)

4minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

+ 120572119896120576119896

(43)

Finally we get 1198781

+ 1198782

+ 1198783

le 0 it follows thatΓ119896

(119909119896

119909lowast

)119896isin119873

is a nonincreasing sequence By (21) weknow that Γ

119896

(119909119896

119909lowast

)119896isin119873

is bounded below by 0 HenceΓ119896

(119909119896

119909lowast

)119896isin119873

converges in 119877 and 119909119896

119896isin119873

is boundedPassing to the limit in (42) then 119909

119896

minus 119909119896+1

2

rarr 0 (119896 rarr

infin)

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

(2∘

) If (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119889)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120573 minus 119860

119896

4120583minus

120572119896119888

120583minus

119871

2]10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(44)

Similarly to (1∘

) we can obtain the result(2) When 120583119889 lt 119888

120572119896(119903 minus 119889)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+ 120572119896(119903 minus

119888

120583)10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge 120572119896(119903 minus

119888

120583)

10038171003817100381710038171003817119909119896+1

minus 119909lowast10038171003817100381710038171003817

2

+10038171003817100381710038171003817119909119896

minus 119909lowast10038171003817100381710038171003817

2

ge120572119896(119903 minus 119888120583)

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(45)

Likewise we also discuss in two cases(1∘

) When (120573 minus 119860119896)2120583 ge 120572

119896(119903 minus 119888120583)2

1198781+ 1198782+ 1198783

le minus(120573

2minus

120572119896119889

120583)10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

2

minus [120572119896(119903 minus 119888120583)

4minus

120572119896119888

120583minus

119871

2]

times10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 120572119896120576119896

(46)

Similarly to the proof of (1) we omit the process and getthe conclusion

(2∘

) When (120573 minus 119860119896)2120583 lt 120572

119896(119903 minus 119888120583)2

Similar to the proof of (1) we omit the process and get theconclusion

Theorem 16 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of equilibrium problem

Proof Let 119909lowast be the limiting point of 119909119896119896isin119873

and denote by119909119896

119896isin119870sub119873

some subsequence converging to 119909lowast According to

119909119896+

= argmin119910isin119862

120572119896119891 (119909119896

119910) + ℎ (119910) minus ℎ (119909119896

) minus ⟨nablaℎ (119909119896

) 119910⟩

= argmin119910isin119862

120572119896119891 (119909119896

119910)

(47)

we obtain

0 le 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

le 119891 (119909119896

119910) minus1

120583119891 (119909119896

119909119896+

) lArrrArr

0 le (120583 minus 1) 119891 (119909119896

119910) + 119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

= (120583 minus 1) 119891 (119909119896

119910) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119910) forall119910 isin 119862

(48)

In particular we set 119910 = 119909119896+1 then

0 le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

+ 119891 (119909119896+

119909119896+1

)

le (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ 11988910038171003817100381710038171003817119909119896+1

minus 119909119896+

10038171003817100381710038171003817

2

minus120573

2120572119896

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) + 11988810038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+119888

2

10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

= (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

+ (119889 minus120573

2120572119896

+119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

(49)

That is

(120573

2120572119896

minus 119889 minus119888

2)10038171003817100381710038171003817119909119896+

minus 119909119896+1

10038171003817100381710038171003817

2

le (120583 minus 1) 119891 (119909119896

119909119896+1

) +119888

2

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

(50)

Passing to the limit in (50) as 119896 rarr infin then

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 010038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

2

997888rarr 0

(51)

At the same time 120573 gt 120572119896(2119889 + 119888) so 119909

119896+

minus 119909119896+1

2

rarr 0From 119909

119896

minus 119909119896+1

2

rarr 0 we have 119909119896+

minus 119909119896

2

rarr 0Moreover 119909lowast minus 119909

119896

2

rarr 0 we get 119909119896+ minus 119909lowast

2

rarr 0

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Due to 119891(119909119896

119909119896+

) le 119891(119909119896

119909119896+

) at 119909119896 and 120583119891(119909119896

119909119896+

) ge

119891(119909119896

119909119896+

) then

1

120583119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) (52)

For all 119896 isin 119870 when 119896 rarr infin we have

119909119896

997888rarr 119909lowast

119909119896+

997888rarr 119909lowast

(53)

In addition 119891 is continuous we have

119891 (119909119896

119909119896+

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (119896 997888rarr infin) (54)

Hence

119891 (119909119896

119909119896+

) 997888rarr 0 (119896 997888rarr infin) (55)

Since 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus

⟨nablaℎ(119909119896

) 119910⟩ we have

0 isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) minus nablaℎ (119909119896

) + nablaℎ (119909119896+

) (56)

That is

nablaℎ (119909119896

) minus nablaℎ (119909119896+

) isin 120597 120572119896119891 (119909119896

sdot) + 120595119862 (119909119896+

) (57)

where 120593119862denotes the indicate function of the set119862 Using the

definition of subdifferential we get

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge1

120572119896

⟨nablaℎ (119909119896

) minus nablaℎ (119909119896+

) 119910 minus 119909119896+

⟩ forall119910 isin 119862

(58)

Applying the Cauchy-Schwarz inequality and the proper-ties 119891(119909

119896

sdot) le 119891(119909119896

sdot) and that nablaℎ is Lipschitz continuous on119862 with constant 119871 we have for all 119910 isin 119862

119891 (119909119896

119910) minus 119891 (119909119896

119909119896+

)

ge minus1

120572119896

10038171003817100381710038171003817nablaℎ (119909

119896

) minus nablaℎ (119909119896+

)10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817

ge minus119871

120572119896

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817 forall119910 isin 119862

(59)

Take the limit about 119896 isin 119873 we deduce

119891 (119909lowast

119910) ge 0 forall119910 isin 119862 (60)

Because 119891 is continuous when 119896 rarr infin

119891 (119909119896

119910) 997888rarr 119891 (119909lowast

119910) 10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817997888rarr 0

10038171003817100381710038171003817119910 minus 119909119896+

10038171003817100381710038171003817997888rarr

1003817100381710038171003817119910 minus 119909lowast1003817100381710038171003817

(61)

We finish the proof

For practical implementation it is necessary to give astopping criterion

Definition 17 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (EP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(119891 (119909lowast

sdot) + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (62)

Proposition 18 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(119891(119909

119896

sdot) + 120595119888)(119909119896

)

Theorem 19 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 15 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Proof Here we still discuss in two cases(1) Under perfect foresightUnder perfect foresight it is easy to get 119909119896+ = 119909

119896+1Since 119909

119896

119896isin119873

is infinite it follows from Theorem 16 thatthe sequence converges to some solution 119909

lowast of problem (EP)On the other hand for all 119896 we have

0 le1003817100381710038171003817100381712057411989610038171003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817100381710038171003817

nablaℎ (119909119896

) minus nablaℎ (119909119896+

)

120572119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

le119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817=

119871

120572

10038171003817100381710038171003817119909119896

minus 119909119896+1

10038171003817100381710038171003817

(63)

Because nablaℎ is Lipschitz-continuous with constant 119871 120572119896ge

120572 gt 0Since lim

119896rarrinfin119909119896

minus 119909119896+1

= 0 we obtain that thesequence 120574

119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816

le1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817=

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+1

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0 as 119896 997888rarr infin

(64)

Finally by continuity of 119891 so that when 119896 rarr infin

119891 (119909119896

119909119896+1

) 997888rarr 119891 (119909lowast

119909lowast

) = 0 (65)

119891(119909119896

119909119896+1

) = 119891(119909119896

119909119896+1

) rarr 0 (119896 rarr infin) that is 120575119896 rarr 0

(119896 rarr infin)(2) Under imperfect foresightWe derive that lim

119896rarrinfin119909119896

minus 119909119896+

= 0 in the process ofprovingTheorem 16

Hence the sequence 120574119896

119896isin119873

converges to zeroMoreover

10038161003816100381610038161003816⟨120574119896

119909119896+

minus 119909119896

⟩10038161003816100381610038161003816le

1003817100381710038171003817100381712057411989610038171003817100381710038171003817

10038171003817100381710038171003817119909119896+

minus 11990911989610038171003817100381710038171003817

997904rArr ⟨120574119896

119909119896+

minus 119909119896

⟩ 997888rarr 0

(66)

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

because 119891 is continuous so when 119896 rarr infin 119891(119909119896

119909119896+

) rarr

119891(119909lowast

119909lowast

) = 0At the same time

119891 (119909119896

119909119896+

) le 119891 (119909119896

119909119896+

) le 120583119891 (119909119896

119909119896+

) (67)

Hence 119891(119909119896

119909119896+

) rarr 0 that is 120575119896 rarr 0 (119896 rarr infin)

Next we give the predictor-corrector algorithm about the(EP) with stopping criterion

Algorithm 20 (the predictor-corrector algorithms for (EP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Finding a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) at 119909 bypredictor-corrector method let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(68)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

4 Application to VariationalInequality Problems

Variational inequalities theory which was introduced byStampacchia [21] provides us with a simple direct naturalgeneral efficient and unified framework to study a wideclass of problems arising in pure and applied sciences It hasbeen extended and generalized in several directions usinginnovative and novel techniques for studying a wide class ofequilibrium problems arising in financial economics trans-portation elasticity and optimization During the last threedecades there has been considerable activity in the develop-ment for solving variational inequalities For the applicationsphysical formulation numerical methods and other aspectsof variational inequalities see [21ndash27] and the referencestherein

Let 119865 119862 rarr 119862lowast be a given mapping variational ine-

quality problems are as follows

finding an 119909lowast

isin 119862

st ⟨119865 (119909lowast

) 119910 minus 119909lowast

⟩ ge 0 forall119910 isin 119862

(VIP)

Wedenote119891(119909 119910) = ⟨119865(119909) 119910minus119909⟩ then the problem (EP)is equivalent to the problem (VIP)

Similarly to Assumption 14 we have the following

Assumption 21 Suppose that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

In the same way we consider the following two casesperfect foresight and unperfect foresight cases

First case is under perfect foresightSimilar to Propositions 12 and 13 we have the following

Proposition 22 Assume that there exist 119903 119888 119889 gt 0 and anonnegative function 119892 119862 times 119862 rarr 119877 such that for all119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119892(119910 119909)(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119892(119909 119910) + 119889119911 minus 119910

2

If the sequence 120572119896119896isin119873

is nonincreasing and the 120572119896le 1205731205832119889

for all 119896 isin 119873 and if 119888120583 le 120583 le 1 then the sequence 119909119896119896isin119873

generated by the predictor-corrector algorithm is bounded andlim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Proposition 23 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

Second case is under imperfect foresight

Assumption 24 Assume that there exist 119903 119888 119889 gt 0 and 119903 ge

max 119889 119888120583 for all 119909 119910 119911 isin 119862

(i) ⟨119865(119909) 119910 minus 119909⟩ ge 0 rArr ⟨119865(119910) 119909 minus 119910⟩ le minus119903119909 minus 1199102

(ii) ⟨119865(119910) minus 119865(119911) 119909 minus 119910⟩ le 119888119910 minus 1199112

+ 119889119909 minus 1199102

We denote

119860119896=

120572119896119889

120583+

120583

2

119861119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583

119862119896=

120572119896119889

120583+

120583

2+

119889 minus 119903

4+

119888

2120583

119863119896= 2120583120572

119896119888 + 2120572

119896119889 + 119871120583 + 4119888120572

119896

119864119896= min

120573 minus 119861119896

120572119896

+119903120583 minus 119888

2120583120573 minus 119862

119896

120572119896

(1 + 8120583) 120573 minus 119863119896minus 8120583119861

119896

8120583120572119896

(69)

Theorem 25 Suppose that 119865(119909) satisfies Assumption 24 and120573 ge max119860

119896 119861119896 119862119896 119863119896 120576119896

le 119864119896119909119896

minus 119909119896+

2

then thesequence 119909119896

119896isin119873generated by the predictor-corrector methods

is bounded and lim119896rarrinfin

119909119896

minus 119909119896+1

= 0

Theorem 26 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873If the sequence 119909

119896119896isin119873

generated by the predictor-correctoralgorithm is bounded and lim

119896rarrinfin119909119896

minus119909119896+1

= 0 then everylimit point of 119909119896

119896isin119873is a solution of (VIP)

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Similar toTheorems 15 and 16 we can proveTheorems 25and 26 Here we will omit their details

Moreover we can also give a stopping criterion

Definition 27 Let Δ ge 0 A point 119909lowast is called a Δ-stationarypoint of problem (VIP) if 119909lowast isin 119862 and

exist120574 isin 120597Δ(⟨119865 (119909

lowast

) sdot minus 119909lowast

⟩ + 120593119862) (119909lowast

) st 10038171003817100381710038171205741003817100381710038171003817 le Δ (70)

Proposition 28 (see [20]) Let 119909119896+

= argmin119910isin119862

120572119896119891(119909119896

119910) + ℎ(119910) minus ℎ(119909119896

) minus ⟨nablaℎ(119909119896

) 119910 minus 119909119896

⟩ 120574119896 = (1120572119896)[nablaℎ(119909

119896

) minus

nablaℎ(119909119896+

)] = (1120572119896)[nablaℎ(119909

119896

) minus nablaℎ(119909119896+1

)] 120575119896 = ⟨120574119894

119896 119909119896+

minus 119909119896

⟩ minus

119891(119909119896

119909119896+

)Then 120575

119896

ge 0 and 120574119896

isin 120597120575119896(⟨119865(119909

119896

) sdot minus 119909119896

⟩ + 120595119888)(119909119896

)

Theorem 29 Assume that 120572119896

ge 120572 gt 0 for all 119896 isin 119873

and that the assumptions of Theorem 25 hold Let 119909119896

119896isin119873

be generated by the predictor-corrector algorithm then thesequences 120574119896

119896isin119873and 120575

119896

119896isin119873

converge to zero

Likewise we omit the proofFinally we have the predictor-corrector algorithm for

variational inequalities problems as follows

Algorithm 30 (the predictor-corrector algorithms for (VIP))Let 120572119896ge 120572 gt 0 120576

119896gt 0 for all 119896 isin 119873

Step 1 Let 119896 = 0 1199090 isin 119862 and 120575 gt 0

Step 2 Find a 120583-approximation 119891(119909 sdot) of 119891(119909 sdot) = ⟨119865(119909) sdot minus

119909⟩ at 119909 by predictor-corrector method Let

119909119896+

= min119910isin119862

120572119896119891 (119909119896

119910) + 119867(119909119896

119910)

120576119896le 119864119870

10038171003817100381710038171003817119909119896

minus 119909119896+

10038171003817100381710038171003817

2

119909119896+1

isin 120576119896minusmin119910isin119862

120572119896119891 (119909119896+

119910) + 119867(119909119896+

119910)

(71)

Step 3 If 119909119896 minus 119909119896+1

2

lt 120575 then stop otherwise put 119896 = 119896 + 1

and go to Step 2

5 Conclusions

In this paper wemainly present a predictor-correctormethodfor solving nonsmooth convex equilibrium problems basedon the auxiliary problem principle In the main algorithmeach stage of computation requires two proximal steps Onestep serves to predict the next point the other helps tocorrect the new prediction This method can operate well inpractice At the same time we present convergence analysisunder perfect foresight and imperfect one In particularwe introduce a stopping criterion which gives rise to Δ-stationary points Moreover we apply this algorithm forsolving the particular case variational inequalities

For further work the need can be anticipated here weonly give the conceptual algorithmic framework to solve thisclass of (EP) we will continue to study its rapidly convergentexecutable algorithm andwe will consider how to use bundle

techniques to approximate proximal points and other relatedquantities Moreover we will strive to extend the nonsmoothconvex equilibrium problems to nonconvex cases where itsrelated theory will be researched in later papers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] E Blum andW Oettli ldquoFrom optimization and variational ine-qualities to equilibrium problemsrdquo The Mathematics Studentvol 63 no 1ndash4 pp 123ndash145 1994

[2] F Giannessi and A Maugeri Variational Inequalities and Net-work Equilibrium Problems Plenum Press New York NY USA1995

[3] F Giannessi A Maugeri and P M Pardalos Equilibrium Prob-lems Nonsmooth Optimization and Variational Inequality Mod-els Kluwer Academic Dordrecht The Netherlands 2001

[4] M A Noor and W Oettli ldquoOn general nonlinear complemen-tarity problems and quasi-equilibriardquo Le Matematiche vol 49no 2 pp 313ndash331 (1995) 1994

[5] J Eckstein ldquoNonlinear proximal point algorithms using Breg-man functions with applications to convex programmingrdquoMathematics of Operations Research vol 18 no 1 pp 202ndash2261993

[6] RGlowinski J L Lions andR TremolieresNumerical Analysisof Variational Inequalities North-Holland Amsterdam TheNetherlands 1981

[7] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities and Their Applications Academic PressLondon UK 1980

[8] M A Noor ldquoSome recent advances in variational inequalitiesmdashpart 1 basic conceptsrdquoNewZealand Journal ofMathematics vol26 no 1 pp 53ndash80 1997

[9] M A Noor ldquoSome recent advances in variational inequalitiesPart 2 other conceptsrdquo New Zealand Journal of Mathematicsvol 26 pp 229ndash255 1997

[10] M PatrikssonNonlinear Programming andVariational Inequal-ity Problems AUnifiedApproach KluwerAcademicDordrechtThe Netherlands 1999

[11] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

[12] D L Zhu and PMarcotte ldquoCo-coercivity and its role in the con-vergence of iterative schemes for solving variational inequal-itiesrdquo SIAM Journal on Optimization vol 6 no 3 pp 714ndash7261996

[13] A N Iusem and W Sosa ldquoIterative algorithms for equilibriumproblemsrdquo Optimization vol 52 no 3 pp 301ndash316 2003

[14] H Brezis L Nirenberg and G Stampacchia ldquoA remark on KyFanrsquos minimax principlerdquo Bollettino della Unione MatematicaItaliana vol 6 pp 293ndash300 1972

[15] A N Iusem andW Sosa ldquoNew existence results for equilibriumproblemsrdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 52 no 2 pp 621ndash635 2003

[16] M A Noor ldquoExtragradient methods for pseudomonotone var-iational inequalitiesrdquo Journal of OptimizationTheory and Appli-cations vol 117 no 3 pp 475ndash488 2003

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

[17] M A Noor M Akhter and K I Noor ldquoInertial proximalmethod for mixed quasi variational inequalitiesrdquo NonlinearFunctional Analysis and Applications vol 8 no 3 pp 489ndash4962003

[18] N El Farouq ldquoPseudomonotone variational inequalities con-vergence of proximal methodsrdquo Journal of OptimizationTheoryand Applications vol 109 no 2 pp 311ndash326 2001

[19] S D Flam and A S Antipin ldquoEquilibrium programming usingproximal-like algorithmsrdquoMathematical Programming vol 78no 1 pp 29ndash41 1997

[20] T T V Nguyen J J Strodiot and V H Nguyen ldquoA bundlemethod for solving equilibrium problemsrdquo Mathematical Pro-gramming B vol 116 no 1-2 pp 529ndash552 2009

[21] G Stampacchia ldquoFormes bilineaires coercitives sur les ensem-bles convexesrdquo Comptes Rendus de lrsquoAcademie des Sciences vol258 pp 4413ndash4416 1964

[22] M Bounkhel L Tadj and A Hamdi ldquoIterative schemes to solvenonconvex variational problemsrdquo Journal of Inequalities in Pureand Applied Mathematics vol 4 pp 1ndash14 2003

[23] F H Clarke Y S Ledyaev and P RWolenskiNonsmooth Ana-lysis and Control Theory Springer Berlin Germany 1998

[24] D Kinderlehrer and G Stampacchia An Introduction to Vari-ational Inequalities andTheir Applications SIAM PhiladelphiaPa USA 2000

[25] M A Noor ldquoGeneral variational inequalitiesrdquo Applied Mathe-matics Letters vol 1 no 2 pp 119ndash122 1988

[26] M A Noor ldquoSome developments in general variational ine-qualitiesrdquo Applied Mathematics and Computation vol 152 no1 pp 199ndash277 2004

[27] M A Noor Principles of Variational Inequalities Lap-LambertAcademic Saarbrucken Germany 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of