Research Article Novel Interior Point Algorithms for Solving Nonlinear Convex ...downloads.hindawi.com/journals/aor/2015/487271.pdf · 2019-07-31 · Research Article Novel Interior

Research ArticleNovel Interior Point Algorithms for Solving Nonlinear ConvexOptimization Problems

Sakineh Tahmasebzadeh1 Hamidreza Navidi1 and Alaeddin Malek2

1Department of Applied Mathematics Shahed University PO Box 3319118651 Tehran Iran2Department of Mathematics Tarbiat Modares University PO Box 14115-175 Tehran Iran

Correspondence should be addressed to Hamidreza Navidi navidishahedacir

Received 13 April 2015 Revised 26 August 2015 Accepted 30 August 2015

Academic Editor Ching-Jong Liao

Copyright copy 2015 Sakineh Tahmasebzadeh et alThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This paper proposes three numerical algorithms based on Karmarkarrsquos interior point technique for solving nonlinear convexprogramming problems subject to linear constraints The first algorithm uses the Karmarkar idea and linearization of the objectivefunctionThe second and third algorithms are modification of the first algorithm using the Schrijver andMalek-Naseri approachesrespectively These three novel schemes are tested against the algorithm of Kebiche-Keraghel-Yassine (KKY) It is shown that thesethree novel algorithms are more efficient and converge to the correct optimal solution while the KKY algorithm fails in some casesNumerical results are given to illustrate the performance of the proposed algorithms

1 Introduction

The simplex method has never had any serious competitionuntil 1984 when Karmarkar proposed a new-polynomial-time algorithm in order to solve linear programming prob-lems especially for solving large scale problems [1 2] Polyno-mial complexity of Karmarkarrsquos algorithm has an advantagein comparison with exponential complexity of the simplexalgorithm [3ndash5] The improvement of the Karmarkarrsquos algo-rithm by Schrijver [6 7] resulted in less number of iterationscompared to Karmarkarrsquos method In 2004Malek andNaseri[8] proposed a modified technique based on the interiorpoint algorithm to solve linear programming problems moreefficiently

After the appearance of Karmarkarrsquos algorithm for solvinglinear programming problems the researchers developedthe algorithm to solve the convex quadratic programmingproblem (eg see [9]) Considering the success of the interiormethods for solving linear programming problems [10 11]the researchers used the linearization methods to solveconvex nonlinear programming problems In 2007 Kebbicheet al [12] introduced a projective interior point method tosolve more general problems with nonlinear convex objectivefunctionThe objective of this paper is to propose an optimal

step length in each iteration in combinationwith Karmarkarrsquosalgorithm in order to decrease the nonlinear objective func-tion as fast as possible

In Section 2 extension of Karmarkarrsquos algorithm for non-linear programming problem is considered In Section 3 byconsidering the technique associated with KKYrsquos algorithma modified algorithm is presented In Section 4 the conver-gency of modified algorithm in Section 3 is provedThen thisalgorithm is combined with Schrijverrsquos and Malek-Naserirsquosalgorithms successfully In Section 5 numerical results areillustrated for KKY algorithm and the three suggested modi-fied algorithms are compared to each other

2 Extension of Karmarkarrsquos Algorithm forNonlinear Problems

Consider the following nonlinear optimization problem

min 119891 (119909)

st 119860119909 = 119887

119909 ge 0

(1)

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2015 Article ID 487271 7 pageshttpdxdoiorg1011552015487271

2 Advances in Operations Research

where 119891 119877119899 rarr 119877 is a nonlinear convex and differentiablefunction 119860 is 119898 times 119899 matrix of rank 119898 and 119887 is 119898 times 1 Thestarting point 1199090 gt 0 is chosen to be feasible 119868 is the identitymatrix with order (119899+1) and 119890119879

119899+1is a row vector of 119899+1 ones

Assume 119910 = (1199101 1199102 119910

119899+1)119879 = (119910[119899] 119910

119899+1)119879 the

simplex 119878 = 119910 isin 119877119899+1 | 119890119879119899+1

119910 = 1 119910 ge 0 and the projectivetransformation of Karmarkar T

119896defined by T

119896 119877119899 rarr 119878

such that T119896(119909) = 119910 Consider 119892(119910) = 119910

119899+1[119891(Tminus1119896+1

(119910)) minus

119891(Tminus1119896

(119910))] where 119892 119877119899+1 rarr 119877 is a nonlinear convexand differentiable function and the optimal value of 119892 iszero Using the linearization process to the function 119892 inneighborhood of the ball of center 1199100 as 119892(119910) = 119892(1199100) +

nabla119892(1199100)(119910 minus 1199100) for 119910 isin 119910 isin 119877119899+1 | 119910 minus 1199100 le 120573 alongwith Karmarkar projective transformation [12] we concludethat Problem (1) is equivalent to

min nabla119892 (1199100)119879

119910

st 119860119896119910 = 0

119890119879119899+1

119910 = 1

10038171003817100381710038171003817119910 minus 119910010038171003817100381710038171003817 le 120573

(2)

As a result the optimal solution of the preceding problemlies along the negative projection of the gradient 119862

119901and is

given as 119910119896+1 = 1199100 minus 120573(119862119901119862119901) where 1199100 = (1(119899 +

1) 1(119899 + 1))119879 is the center of the simplex 119878 119862119901is the

projected gradient which can be shown to be 119862119901

= [119868 minus

119875119879(119875119875119879)minus1119875]nabla119892(1199100) where 119875 = (119860119896

119890119879

119899+1

) 119860119896

= [119860119863119896 minus119887]

119863119896

= diag119909119896 and 119896 is the number of iterations Theselection of 120573 as a step length is crucial to enhance theefficiency of the algorithm

The function 119892 on 119884 = 119910 isin 119877119899+1 | 119860119896119910 = 0 119910 isin 119878 is

convex since the function 119891 on 119883 = 119909 isin 119877119899 | 119860119909 = 119887 119909 ge0 is convex The optimal solution of Problem (2) is given by119910119896+1 = 1199100 minus 120573(119862

119901119862119901) [12]

Consider the algorithm of Kebiche-Keraghel-Yassine(KKY) for solving Problem (2)

KKY Algorithm Let 120598 gt 0 be a given tolerance and let 1199090 gt 0

Step 1 Compute 1199100 = (1(119899 + 1) 1(119899 + 1))119879 and 120573 =(119899 + 1)3(119899 + 2) Put 119896 = 0

Step 2 Build 119863119896

= diag119909119896 119860119896

= [119860119863119896 minus119887] and 119875 =

(119860119896

119890119879

119899+1

)Compute 119862

119901= [119868 minus 119875119879(119875119875119879)minus1119875]nabla119892(1199100) 119910119896+1 = 1199100 minus

120573(119862119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1

Step 3 While 119891(119909119896+1) minus 119891(119909119896) gt 120598 let 119896 = 119896 + 1 and go toStep 2

However as we can see in the following two examples theKKY algorithm does not work for every nonlinear problem

Example 1 Consider the quadratic convex problem

min 119891 (119909) = 11990921+ 11990922minus 21199091minus 41199092

st 1199091+ 41199092+ 1199093= 5

21199091+ 31199092+ 1199094= 6

1199091 1199092 1199093 1199094ge 0

(3)

For tolerance 120598 = 10minus8 after 119896 = 161 iterations for KKYalgorithm we have

119909161 = [000000000 200005934 minus300008255minus000000000]119879 and 119891(119909161) = minus39999999964 Sincethe condition 119891(119909162) minus 119891(119909161) = 447 times 10minus9 lt 10minus8

holds the algorithm stopped whereas this solution isnot feasible

Example 2 Consider the nonlinear convex problem

min 119891 (119909)

=3

2(11990921+ 11990922) + 2 (1199092

3+ 11990924) minus ln (119909

11199094)

minus 311990911199092+ 411990931199094minus 21199091minus 31199094

st 1199091+

1

41199094minus 81199095minus 1199096+ 91199097= 0

1199092+

1

21199094minus 12119909

5minus

1

21199096+ 31199097= 0

1199093+ 1199096= 1

1199091 1199092 119909

7ge 0

(4)

For 120598 = 10minus8 120573 = 827 isin [027 036] and119896 = 1 iteration using KKY algorithm the solution 1199091 =

[minus52199 minus18871 04463 03791 minus00087 05536 06232]119879

is calculated This solution is not feasible

Here two difficulties arise First 119909119896 must stay feasiblein each iteration Second in the moderate time the requiredtolerance must be satisfied To overcome these difficulties wemodify the KKY algorithm in the next section

3 Modified Algorithm

31Modifications In theKKY [12] algorithm120573 isin [027 036]As it is observed in Examples 1 and 2 for these values of 120573the KKY algorithm may produce infeasible solution In thestandard Karmarkar [1 3 4] instead of 120573 120572119903 = ((119899 + 1)

3(119899 + 2))(1radic(119899 + 1)(119899 + 2)) is used Hence the solutionin each iteration remains feasible In this case the optimalsolution for Problem (2) is given by 119910119896+1 = 1199100 minus 120573(119862

119901119862119901)

where 120573 = 120572119903 Applying KKY algorithm with 120573 = 120572119903 to theproblem in Example 2 it gives the feasible solution 11990980 =[07517262 00000174 00000056 10209196 0000874209999943 00000035]119879 119891(11990980) = minus13692774913 and119891(11990981) = minus13692882373 The algorithm stops after 119896 = 80

Advances in Operations Research 3

iterations since 119891(11990981) minus 119891(11990980) = minus10746 times 10minus5 while thesuitable accuracy 120598 = 10minus8 is not reached In each iterationlinear search method is used to find 0 lt 120582 le 1 such that119891(119909119896+1) le 119891(119909119896) where the suitable tolerance satisfies Thusone may write Problem (2) in the following form

min nabla119892 (1199100)119879

119910

st 119860119896119910 = 0

119890119879119899+1

119910 = 1

10038171003817100381710038171003817119910 minus 119910010038171003817100381710038171003817 le 120582120572119903

(5)

Lemma 3 The optimal solution for Problem (5) is given by119910119896+1 = 1199100 minus 120582120572119903(119862

119901119862119901)

Proof We put 119909 = 119910minus1199100 and then we have 119875119909 = (119860119896

119890119879

119899+1

) (119910 minus

1199100) = 0 and Problem (5) is equivalent to

min nabla119892 (1199100)119879

119909

st 119875119909 = 0

1199092 le 120582212057221199032

(6)

119909lowast is a solution of Problem (6) if and only if there exist 120596 isin

119877119898+1 and 120583 ge 0 such that

nabla119892 (1199100) + 119875119879120596 + 120583119909lowast = 0 (7)

Multiplying both sides of (7) by 119875 and since 119875119909lowast = 0 weget 119875nabla119892(1199100) + 119875119875119879120596 = 0 Then 120596 = minus(119875119875119879)minus1(119875nabla119892(1199100))by substituting 120596 in (7) we find

119909lowast = minus1

120583[119868 minus 119875119879 (119875119875119879)

minus1

119875]nabla119892 (1199100) (8)

By assuming 119862119875= [119868 minus 119875119879(119875119875119879)minus1119875]nabla119892(1199100) we have

1003817100381710038171003817119909lowast1003817100381710038171003817 =

1

120583

10038171003817100381710038171198621198751003817100381710038171003817 = 120582120572119903

997904rArr1

120583=

1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817

997904rArr 119909lowast = minus120582120572119903119862119875

10038171003817100381710038171198621198751003817100381710038171003817

(9)

And we have 119910119896+1 = 119910lowast = 1199100 +119909lowast = 1199100 minus120582120572119903(119862119875119862119875)

Note that here we have proposed the algorithm similarto KKY where 120573 = 120582120572119903 This modified algorithm has theadvantage that can find the feasible approximate solution inthe suitable tolerance

32 Modified KKY Algorithm (MKKY) Let 120598 gt 0 be a giventolerance and 1199090 is a strictly feasible point

Step 1 Compute 1199100 = (1(119899 + 1) 1(119899 + 1))119879 119903 =

1radic(119899 + 1)(119899 + 2) and 120572 = (119899 + 1)3(119899 + 2) Put 120582 = 1 and119896 = 0

Step 2 Build 119863119896

= diag119909119896 119860119896

= [119860119863119896 minus119887] and 119875 =

(119860119896

119890119879

119899+1

)Compute 119862


120582120572119903(119862119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1

Step 3 While 119891(119909119896+1) minus 119891(119909119896) gt 120598 put 119896 = 119896 + 1Build119863

119896= diag119909119896 119860

119896= [119860119863

119896 minus119887] and 119875 = (

119860119896

119890119879

119899+1

)Compute 119862

119901= [119868 minus119875119879(119875119875119879)minus1119875]nabla119892(1199100) and 119910119896+1 = 1199100 minus

120582120572119903(119862119901119862119901)

Then 119909119896+1 = 119879minus1119896

(119910119896+1) = 119863119896119910119896+1[119899]119910119870

119899+1

Let 119896 = 119896 + 1 and go to Step 3

Step 4 While 119891(119909119896+1) gt 119891(119909119896) compute 120582 = (12)120582 119910119896+1 =1199100 minus 120582120572119903(119862

119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1

Step 5 Let 119896 = 119896 + 1 and go to Step 3

4 Convergence for ModifiedAlgorithm (MKKY)

In order to establish the convergence of the modified algo-rithm we introduce a potential function associated withProblem (1) defined by

120601 (119909) = (119899 + 1) ln (119891 (119909) minus 119911lowast) minus119899

sum119894=1

ln (119909119894) (10)

where 119911lowast is the optimal value of the objective function

Lemma 4 If 119910119896+1 is the optimal solution of Problem (5) then119892(119910119896+1) lt 119892(1199100)

Proof Since 119892(119910119896+1) = 119892(1199100)+nabla119892(1199100)119879(119910119896+1minus1199100) and119910119896+1 =

1199100 minus 120582120572119903(119862119875119862119875) then

119892 (119910119896+1) minus 119892 (1199100) = nabla119892 (1199100)119879

(minus120582120572119903119862119875

10038171003817100381710038171198621198751003817100381710038171003817)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817(nabla119892 (1199100) 119862

119875)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817

100381710038171003817100381711986211987510038171003817100381710038172

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817 lt 0

(11)

Thus 119892(119910119896+1) lt 119892(1199100) and a reduction is obtained in eachiteration

Lemma 5 If 119910119896+1 is the optimal solution of Problem (5) then

119892 (119910119896+1) le (1 minus120582120572

119899 + 1)119892 (1199100) (12)

Proof Let 119909lowast be the optimal solution of Problem (1) we canwrite 119909lowast = Tminus1

119896(119910lowast) 119861(1199100 120582120572119903) is a ball of center 1199100 with

radius 120582120572119903 There are two cases


(i) If 119910lowast isin 119884 cap 119861(1199100 120582120572119903) then

0 le 119892 (119910119896+1) le 119892 (119910lowast) = 0 lt (1 minus120582120572

119899 + 1)119892 (1199100) (13)

and Lemma 5 holds(ii) If 119910lowast notin 119884cap119861(1199100 120582120572119903) since 119884 is convex intersection

point of the boundary of 119884 cap 119861(1199100 120582120572119903) and the linesegment between 1199100 and 119910lowast should be feasible forProblem (5) Let 119910119896+1 be the intersection point then119910119896+1 satisfies 119910119896+1 = 120579119910lowast + (1minus120579)1199100 for 120579 isin (0 1) and119910119896+1 minus 1199100 = 120582120572119903 Thus

10038171003817100381710038171003817119910119896+1 minus 1199100

10038171003817100381710038171003817 =10038171003817100381710038171003817120579 (119910lowast minus 1199100)

10038171003817100381710038171003817 = 120582120572119903 997904rArr

10038171003817100381710038171003817(119910lowast minus 1199100)

10038171003817100381710038171003817 =120582120572119903

120579

(14)

Hence10038171003817100381710038171003817119910lowast minus 1199100

100381710038171003817100381710038172

=1003817100381710038171003817119910lowast10038171003817100381710038172

+100381710038171003817100381710038171199100100381710038171003817100381710038172

minus 21199100119879

119910lowast (15)

and we have

1003817100381710038171003817119910lowast1003817100381710038171003817 =119899+1

sum119894=1

(119910lowast119894)2

le (119899+1

sum119894=1

119910119894

lowast)

2

= (119890119879119899+1

119910lowast)2

= 1

100381710038171003817100381710038171199100100381710038171003817100381710038172

=1003817100381710038171003817100381710038171003817

1

119899 + 1119890119899+1

1003817100381710038171003817100381710038171003817

2

=1

119899 + 1

1199100119879

119910lowast =1

119899 + 1119890119879119899+1

119910lowast =1

119899 + 1

(16)

Thus

10038171003817100381710038171003817119910lowast minus 1199100

100381710038171003817100381710038172

le 1 +1

119899 + 1minus

2

119899 + 1=

119899

119899 + 1le

119899 + 1

119899 + 2 (17)

From (14) we also have 119910lowast minus 11991002 = (120582120572119903120579)2 therefore

(120582120572119903

120579)2

le119899 + 1

119899 + 2or

120582120572119903

120579le

radic119899 + 1

radic119899 + 2

(18)

Substituting 119903 = 1radic(119899 + 1)(119899 + 2) in the above inequality wehave 120579 ge 120582120572(119899 + 1) Since 119892 is convex and 119892(119910lowast) = 0 then

119892 (119910119896+1) le 120579119892 (119910lowast) + (1 minus 120579) 119892 (1199100)

le (1 minus120582120572

119899 + 1)119892 (1199100)

(19)

Furthermore 119892(119910119896+1) le 119892(119910lowast) then 119892(119910119896+1) le (1 minus 120582120572(119899 +

1))119892(1199100)

Theorem 6 In every iteration of the algorithmMKKY poten-tial function is reduced by a constant value 120575 such that120601(119909119896+1) le 120601(119909119896) minus 120575

Proof Consider

120601 (119909119896+1) minus 120601 (119909119896) = (119899 + 1) ln(119891(119909119896+1) minus 119911lowast

119891 (119909119896) minus 119911lowast)

minus119899

sum119894=1

ln(119909119896+1119894

119909119896119894

)

= (119899 + 1) ln(119892 (119910119896+1)

119892 (1199100))

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le (119899 + 1) ln(1 minus120582120572

119899 + 1)

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le minus120582120572 +12058221205722

2 (1 minus 120582120572)2

(20)

We used the result demonstrated by Karmarkar [1]

minus119899+1

sum119894=1

ln (119910119896+1119894

) le12058221205722

2 (1 minus 120582120572)2 (21)

Then 120601(119909119896+1) le 120601(119909119896) minus 120575 where 120575 = minus120582120572 + 120582212057222(1 minus 120582120572)2If 120572 = (119899 + 1)3(119899 + 2) then 120601(119909119896+1) le 120601(119909119896) minus 02Therefore 120601(119909119896+1) le 120601(1199090) minus 02(119896 + 1)

Theorem 7 If 1199090 is the feasible solution of Problem (1) and119909lowast is the optimal solution with optimal value 119911lowast then one hasfollowing assumptions

(1) 1199090 ge 2minus119871119890119899

(2) 119909lowast le 2119871119890119899 for any feasible solution 119909 one has minus23119871 le

119911lowast le 119891(119909) le 23119871The algorithm MKKY finds an optimal solution after 119874(119899119871)iteration 119871 is the number of bytes

Proof Consider

119891 (119909119896) minus 119911lowast

119891 (1199090) minus 119911lowast= 120574 (119909119896) exp(

120601 (119909119896) minus 120601 (1199090)

119899 + 1) (22)

where

120574 (119909119896) = exp(sum119899

119894=1ln (119909119896119894) minus sum119899

119894=1ln (1199090119894)

119899 + 1) (23)

By assumptions (1) and (2) we have 120574(119909119896) le 22119871 in the feasibleregion then

119891 (119909119896) minus 119911lowast

le 22119897 (119891 (1199090) minus 119911lowast) exp(120601 (119909119896) minus 120601 (1199090)

119899 + 1)

(24)


Table 1 Computed solutions using four different algorithms for Examples 8 9 10 and 11

Examples 119909 KKY MKKY Sch-MKKY MN-MKKY

Example 8

1199091

11176285677 07993104755 07992885091 079846775721199092

08925922545 12006283419 12006478942 120137562451199093

minus00102208222 00000611826 00000635969 000015661831199094

13324440588 03980537916 03979927208 0395765081

Example 9

1199091

minus52199797446 07514629509 07514297188 075142749971199092

minus18871925195 00000032699 00000065967 000000681981199093

04463304312 00000010511 00000021234 000000220201199094

03791651531 10175937272 10172339995 101721015311199095

minus00087312913 00007335551 00007190583 000071809831199096

05536695688 09999989489 09999978766 099999779801199097

06232230772 00000006675 00000013472 00000013940

Example 101199091

00000000025 00000000156 00000000055 000000000801199092

19999999829 19999999858 19999999749 199999995861199093

00000000025 00000000156 00000000055 00000000080

Example 11

1199091

03124560511 07142857707 07142857350 071428574531199092

03438507936 01428572168 01428571684 014285716001199093

00001576431 00000002101 00000000702 000000004981199094

minus14064826279 00000001087 00000000363 00000000258

Table 2 Comparison between four different algorithms for Examples 8 9 10 and 11

Examples Algorithm 119896 119909lowast 119891(119909lowast) Error CPU time (sec)

Example 8

KKY (infeasible) mdash mdash mdash mdash mdashMKKY 549 14962795449 minus71998265727 17344 times 10minus4 1828281

Sch-MKKY 515 14962672544 minus71998196616 18034 times 10minus4 1534063MN-MKKY 187 14958093895 minus71995511207 44888 times 10minus4 835313

Example 9



Example 10

KKY 17 19999999829 minus39999999317 68305 times 10minus8 114844MKKY 80 19999999858 minus39999999433 56661 times 10minus8 465469


Example 11



According to Theorem 6 after 119896 iterations we have 120601(119909119896) minus

120601(1199090) le minus119896120575 Thus

119891 (119909119896) minus 119911lowast le 22119897 (119891 (1199090) minus 119911lowast) exp(minus119896120575

119899 + 1)

le 25119871 exp(minus119896120575

119899 + 1) le 2minus4119871

(25)

Therefore 119891(119909119896) minus 119911lowast le 2minus4119871 for 119896 ge 45119871(119899 + 1)

In the next section MKKY algorithm is combined withthe algorithmof Schrijver [6 7] andMalek-Naserirsquos algorithm[8] to propose novel hybrid algorithms called Sch-MKKY

(Schrijver-Modified Kebiche-Keraghel-Yassine) and MN-MKKY (Malek-Naseri-Modified Kebiche-Keraghel-Yassine)These algorithms are different from MKKY in the use ofoptimal length in each iteration

41 Hybrid Algorithms Let us assume that 119903 and 1199100 inthe modified algorithm are expressed as follows 119903 =radic(119899 + 2)(119899 + 1) and 1199100 = (1 1 1)119879 In the Sch-MKKYand MN-MKKY algorithms choose 120572 = 1(1 + 119903) and 120572 =

1minus1(119899+2)4(1+radic119899(119899 + 1)) respectively It is easy to prove thatthe theorems in Section 4 satisfy Sch-MKKYandMN-MKKYalgorithmsThus with the recent step length the convergenceis guaranteed


minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus10

100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8

minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1log(tolerance)

0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11

Figure 1 The tolerances against number of iterations are compared for MKKY Sch-MKKY and MN-MKKY algorithms for Examples 8 910 and 11

5 Numerical Results

It is observed that the approximate solution from KKYalgorithm is not feasible in some instances (see Examples 89 and 11 in Table 1) All programs have been written withMATLAB 704 for 120598 = 10minus8 as tolerance The computedsolution for KKY Sch-MKKY and MN-MKKY is given inTable 1


min 119891 (119909) = minus21199091minus 61199092+ 11990921minus 211990911199092+ 211990922

st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)

As it is shown the KKY algorithm does not converge tothe correct solution while the computed solution of MKKYand two hybrid algorithms is feasible Example 8 is solvedby using Fmincon from MATLAB 704 and the absolutedifference of the objective functions stated as Error In Table 2number of iterations solution norms optimal values of theobjective function Error and the elapsed time for eachalgorithm have been given

In Table 2 we show that MN-MKKY algorithm is moreefficient than the other algorithms comparing the number ofiterations and elapsed time

In Figure 1 for various tolerances the number of itera-tions is compared for different algorithms It is observed thatthe MN-MKKY algorithm has better performance for eachrequired tolerance

Example 9 Consider Example 2 Table 2 shows the com-puted solution for four different algorithms From Figure 1


it is obvious that MN-MKKY algorithm is the best for thisexample


min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion

Having two ideas in our mind (i) calculation of feasiblesolution in each iteration and (ii) the fact that the objectivefunction value must decrease in each iteration with thefixed desired tolerance this paper proposed three hybridalgorithms for solving nonlinear convex programming prob-lem based on the interior point idea using various 120573rsquos ofKarmarkar Schrijver and Malek-Naseri techniques Thesemethods have better performance than the standard Kar-markar algorithm since in the latter algorithm one may notcheck the feasibility of solution in each iteration

Our numerical simulation shows that the MN-MKKYalgorithm has the best performance among the other algo-rithms This algorithm uses less number of iterations tosolve the general nonlinear optimization problemswith linearconstraints since it uses the step length 120573 of Malek-Naseritype

Conflict of Interests

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

References

[1] N Karmarkar ldquoA new polynomial-time algorithm for linearprogrammingrdquo Combinatorica vol 4 no 4 pp 373ndash395 1984

[2] H Navidi A Malek and P Khosravi ldquoEfficient hybrid algo-rithm for solving large scale constrained linear programmingproblemsrdquo Journal of Applied Sciences vol 9 no 18 pp 3402ndash3406 2009

[3] M S Bazzara J Jarvis and H D Sherali Linear Programmingand Network Flows John Wiley amp Sons New York NY USA1984

[4] A T Hamdy Operations Research Macmillan New York NYUSA 1992

[5] R M R Karp ldquoGeorge Dantzigrsquos impact on the theory ofcomputationrdquo Discrete Optimization vol 5 no 2 pp 174ndash1852008

[6] C Roos T Terlaky and J VialTheory and Algorithms for LinearOptimization Princton University 2001

[7] A Schrijver Theory of Linear and Integer Programming JohnWiley amp Sons New York NY USA 1986

[8] A Malek and R Naseri ldquoA new fast algorithm based onKarmarkarrsquos gradient projected method for solving linear pro-gramming problemrdquoAdvancedModeling andOptimization vol6 no 2 pp 43ndash51 2004

[9] E Tse and Y Ye ldquoAn extension of Karmarkarrsquos projectivealgorithm for convex quadratic programmingrdquo MathematicalProgramming vol 44 no 2 pp 157ndash179 1989

[10] Z Kebbiche and D Benterki ldquoA weighted path-followingmethod for linearly constrained convex programmingrdquo RevueRoumaine de Mathematique Pures et Appliquees vol 57 no 3pp 245ndash256 2012

[11] P Fei and YWang ldquoA primal infeasible interior point algorithmfor linearly constrained convex programmingrdquo Control andCybernetics vol 38 no 3 pp 687ndash704 2009

[12] Z Kebbiche A Keraghel and A Yassine ldquoExtension of aprojective interior pointmethod for linearly constrained convexprogrammingrdquoAppliedMathematics and Computation vol 193no 2 pp 553ndash559 2007

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where 119891 119877119899 rarr 119877 is a nonlinear convex and differentiablefunction 119860 is 119898 times 119899 matrix of rank 119898 and 119887 is 119898 times 1 Thestarting point 1199090 gt 0 is chosen to be feasible 119868 is the identitymatrix with order (119899+1) and 119890119879

119899+1is a row vector of 119899+1 ones

Assume 119910 = (1199101 1199102 119910

119899+1)119879 = (119910[119899] 119910

119899+1)119879 the

simplex 119878 = 119910 isin 119877119899+1 | 119890119879119899+1

119910 = 1 119910 ge 0 and the projectivetransformation of Karmarkar T

119896defined by T

119896 119877119899 rarr 119878

such that T119896(119909) = 119910 Consider 119892(119910) = 119910

119899+1[119891(Tminus1119896+1

(119910)) minus

119891(Tminus1119896

(119910))] where 119892 119877119899+1 rarr 119877 is a nonlinear convexand differentiable function and the optimal value of 119892 iszero Using the linearization process to the function 119892 inneighborhood of the ball of center 1199100 as 119892(119910) = 119892(1199100) +

nabla119892(1199100)(119910 minus 1199100) for 119910 isin 119910 isin 119877119899+1 | 119910 minus 1199100 le 120573 alongwith Karmarkar projective transformation [12] we concludethat Problem (1) is equivalent to

min nabla119892 (1199100)119879

119910

st 119860119896119910 = 0

119890119879119899+1

119910 = 1

10038171003817100381710038171003817119910 minus 119910010038171003817100381710038171003817 le 120573

(2)

As a result the optimal solution of the preceding problemlies along the negative projection of the gradient 119862

119901and is

given as 119910119896+1 = 1199100 minus 120573(119862119901119862119901) where 1199100 = (1(119899 +

1) 1(119899 + 1))119879 is the center of the simplex 119878 119862119901is the

projected gradient which can be shown to be 119862119901

= [119868 minus

119875119879(119875119875119879)minus1119875]nabla119892(1199100) where 119875 = (119860119896

119890119879

119899+1

) 119860119896

= [119860119863119896 minus119887]

119863119896

= diag119909119896 and 119896 is the number of iterations Theselection of 120573 as a step length is crucial to enhance theefficiency of the algorithm

The function 119892 on 119884 = 119910 isin 119877119899+1 | 119860119896119910 = 0 119910 isin 119878 is

convex since the function 119891 on 119883 = 119909 isin 119877119899 | 119860119909 = 119887 119909 ge0 is convex The optimal solution of Problem (2) is given by119910119896+1 = 1199100 minus 120573(119862

119901119862119901) [12]

Consider the algorithm of Kebiche-Keraghel-Yassine(KKY) for solving Problem (2)

KKY Algorithm Let 120598 gt 0 be a given tolerance and let 1199090 gt 0

Step 1 Compute 1199100 = (1(119899 + 1) 1(119899 + 1))119879 and 120573 =(119899 + 1)3(119899 + 2) Put 119896 = 0

Step 2 Build 119863119896

= diag119909119896 119860119896

= [119860119863119896 minus119887] and 119875 =

(119860119896

119890119879

119899+1

)Compute 119862


120573(119862119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1

Step 3 While 119891(119909119896+1) minus 119891(119909119896) gt 120598 let 119896 = 119896 + 1 and go toStep 2

However as we can see in the following two examples theKKY algorithm does not work for every nonlinear problem


min 119891 (119909) = 11990921+ 11990922minus 21199091minus 41199092

st 1199091+ 41199092+ 1199093= 5

21199091+ 31199092+ 1199094= 6

1199091 1199092 1199093 1199094ge 0

(3)

For tolerance 120598 = 10minus8 after 119896 = 161 iterations for KKYalgorithm we have

119909161 = [000000000 200005934 minus300008255minus000000000]119879 and 119891(119909161) = minus39999999964 Sincethe condition 119891(119909162) minus 119891(119909161) = 447 times 10minus9 lt 10minus8

holds the algorithm stopped whereas this solution isnot feasible


min 119891 (119909)

=3

2(11990921+ 11990922) + 2 (1199092

3+ 11990924) minus ln (119909

11199094)

minus 311990911199092+ 411990931199094minus 21199091minus 31199094

st 1199091+

1

41199094minus 81199095minus 1199096+ 91199097= 0

1199092+

1

21199094minus 12119909

5minus

1

21199096+ 31199097= 0

1199093+ 1199096= 1

1199091 1199092 119909

7ge 0

(4)

For 120598 = 10minus8 120573 = 827 isin [027 036] and119896 = 1 iteration using KKY algorithm the solution 1199091 =

[minus52199 minus18871 04463 03791 minus00087 05536 06232]119879

is calculated This solution is not feasible

Here two difficulties arise First 119909119896 must stay feasiblein each iteration Second in the moderate time the requiredtolerance must be satisfied To overcome these difficulties wemodify the KKY algorithm in the next section

3 Modified Algorithm

31Modifications In theKKY [12] algorithm120573 isin [027 036]As it is observed in Examples 1 and 2 for these values of 120573the KKY algorithm may produce infeasible solution In thestandard Karmarkar [1 3 4] instead of 120573 120572119903 = ((119899 + 1)

3(119899 + 2))(1radic(119899 + 1)(119899 + 2)) is used Hence the solutionin each iteration remains feasible In this case the optimalsolution for Problem (2) is given by 119910119896+1 = 1199100 minus 120573(119862

119901119862119901)

where 120573 = 120572119903 Applying KKY algorithm with 120573 = 120572119903 to theproblem in Example 2 it gives the feasible solution 11990980 =[07517262 00000174 00000056 10209196 0000874209999943 00000035]119879 119891(11990980) = minus13692774913 and119891(11990981) = minus13692882373 The algorithm stops after 119896 = 80



min nabla119892 (1199100)119879

119910

st 119860119896119910 = 0

119890119879119899+1

119910 = 1

10038171003817100381710038171003817119910 minus 119910010038171003817100381710038171003817 le 120582120572119903

(5)


119901119862119901)


119890119879

119899+1

) (119910 minus


min nabla119892 (1199100)119879

119909

st 119875119909 = 0

1199092 le 120582212057221199032

(6)


119877119898+1 and 120583 ge 0 such that

nabla119892 (1199100) + 119875119879120596 + 120583119909lowast = 0 (7)



120583[119868 minus 119875119879 (119875119875119879)

minus1

119875]nabla119892 (1199100) (8)


1003817100381710038171003817119909lowast1003817100381710038171003817 =

1

120583

10038171003817100381710038171198621198751003817100381710038171003817 = 120582120572119903

997904rArr1

120583=

1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817


10038171003817100381710038171198621198751003817100381710038171003817

(9)




Step 1 Compute 1199100 = (1(119899 + 1) 1(119899 + 1))119879 119903 =

1radic(119899 + 1)(119899 + 2) and 120572 = (119899 + 1)3(119899 + 2) Put 120582 = 1 and119896 = 0

Step 2 Build 119863119896

= diag119909119896 119860119896

= [119860119863119896 minus119887] and 119875 =

(119860119896

119890119879

119899+1

)Compute 119862


120582120572119903(119862119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1


119896= diag119909119896 119860

119896= [119860119863

119896 minus119887] and 119875 = (

119860119896

119890119879

119899+1

)Compute 119862


120582120572119903(119862119901119862119901)

Then 119909119896+1 = 119879minus1119896

(119910119896+1) = 119863119896119910119896+1[119899]119910119870

119899+1

Let 119896 = 119896 + 1 and go to Step 3


119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1





sum119894=1

ln (119909119894) (10)




1199100 minus 120582120572119903(119862119875119862119875) then

119892 (119910119896+1) minus 119892 (1199100) = nabla119892 (1199100)119879

(minus120582120572119903119862119875

10038171003817100381710038171198621198751003817100381710038171003817)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817(nabla119892 (1199100) 119862

119875)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817

100381710038171003817100381711986211987510038171003817100381710038172

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817 lt 0

(11)



119892 (119910119896+1) le (1 minus120582120572

119899 + 1)119892 (1199100) (12)







119899 + 1)119892 (1199100) (13)



10038171003817100381710038171003817119910119896+1 minus 1199100

10038171003817100381710038171003817 =10038171003817100381710038171003817120579 (119910lowast minus 1199100)

10038171003817100381710038171003817 = 120582120572119903 997904rArr

10038171003817100381710038171003817(119910lowast minus 1199100)

10038171003817100381710038171003817 =120582120572119903

120579

(14)


100381710038171003817100381710038172

=1003817100381710038171003817119910lowast10038171003817100381710038172

+100381710038171003817100381710038171199100100381710038171003817100381710038172

minus 21199100119879

119910lowast (15)

and we have

1003817100381710038171003817119910lowast1003817100381710038171003817 =119899+1

sum119894=1

(119910lowast119894)2

le (119899+1

sum119894=1

119910119894

lowast)

2

= (119890119879119899+1

119910lowast)2

= 1

100381710038171003817100381710038171199100100381710038171003817100381710038172

=1003817100381710038171003817100381710038171003817

1

119899 + 1119890119899+1

1003817100381710038171003817100381710038171003817

2

=1

119899 + 1

1199100119879

119910lowast =1

119899 + 1119890119879119899+1

119910lowast =1

119899 + 1

(16)

Thus

10038171003817100381710038171003817119910lowast minus 1199100

100381710038171003817100381710038172

le 1 +1

119899 + 1minus

2

119899 + 1=

119899

119899 + 1le

119899 + 1

119899 + 2 (17)


(120582120572119903

120579)2

le119899 + 1

119899 + 2or

120582120572119903

120579le

radic119899 + 1

radic119899 + 2

(18)


119892 (119910119896+1) le 120579119892 (119910lowast) + (1 minus 120579) 119892 (1199100)

le (1 minus120582120572

119899 + 1)119892 (1199100)

(19)


1))119892(1199100)


Proof Consider


119891 (119909119896) minus 119911lowast)

minus119899

sum119894=1

ln(119909119896+1119894

119909119896119894

)

= (119899 + 1) ln(119892 (119910119896+1)

119892 (1199100))

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le (119899 + 1) ln(1 minus120582120572

119899 + 1)

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le minus120582120572 +12058221205722

2 (1 minus 120582120572)2

(20)


minus119899+1

sum119894=1

ln (119910119896+1119894

) le12058221205722

2 (1 minus 120582120572)2 (21)



(1) 1199090 ge 2minus119871119890119899



Proof Consider

119891 (119909119896) minus 119911lowast

119891 (1199090) minus 119911lowast= 120574 (119909119896) exp(

120601 (119909119896) minus 120601 (1199090)

119899 + 1) (22)

where

120574 (119909119896) = exp(sum119899

119894=1ln (119909119896119894) minus sum119899

119894=1ln (1199090119894)

119899 + 1) (23)


119891 (119909119896) minus 119911lowast


119899 + 1)

(24)




Example 8

1199091

11176285677 07993104755 07992885091 079846775721199092

08925922545 12006283419 12006478942 120137562451199093

minus00102208222 00000611826 00000635969 000015661831199094

13324440588 03980537916 03979927208 0395765081

Example 9

1199091

minus52199797446 07514629509 07514297188 075142749971199092

minus18871925195 00000032699 00000065967 000000681981199093

04463304312 00000010511 00000021234 000000220201199094

03791651531 10175937272 10172339995 101721015311199095

minus00087312913 00007335551 00007190583 000071809831199096

05536695688 09999989489 09999978766 099999779801199097

06232230772 00000006675 00000013472 00000013940

Example 101199091

00000000025 00000000156 00000000055 000000000801199092

19999999829 19999999858 19999999749 199999995861199093

00000000025 00000000156 00000000055 00000000080

Example 11

1199091

03124560511 07142857707 07142857350 071428574531199092

03438507936 01428572168 01428571684 014285716001199093

00001576431 00000002101 00000000702 000000004981199094

minus14064826279 00000001087 00000000363 00000000258



Example 8



Example 9



Example 10



Example 11




120601(1199090) le minus119896120575 Thus


119899 + 1)

le 25119871 exp(minus119896120575

119899 + 1) le 2minus4119871

(25)








100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8


0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11


5 Numerical Results




st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)








min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min nabla119892 (1199100)119879

119910

st 119860119896119910 = 0

119890119879119899+1

119910 = 1

10038171003817100381710038171003817119910 minus 119910010038171003817100381710038171003817 le 120582120572119903

(5)


119901119862119901)


119890119879

119899+1

) (119910 minus


min nabla119892 (1199100)119879

119909

st 119875119909 = 0

1199092 le 120582212057221199032

(6)


119877119898+1 and 120583 ge 0 such that

nabla119892 (1199100) + 119875119879120596 + 120583119909lowast = 0 (7)



120583[119868 minus 119875119879 (119875119875119879)

minus1

119875]nabla119892 (1199100) (8)


1003817100381710038171003817119909lowast1003817100381710038171003817 =

1

120583

10038171003817100381710038171198621198751003817100381710038171003817 = 120582120572119903

997904rArr1

120583=

1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817


10038171003817100381710038171198621198751003817100381710038171003817

(9)




Step 1 Compute 1199100 = (1(119899 + 1) 1(119899 + 1))119879 119903 =

1radic(119899 + 1)(119899 + 2) and 120572 = (119899 + 1)3(119899 + 2) Put 120582 = 1 and119896 = 0

Step 2 Build 119863119896

= diag119909119896 119860119896

= [119860119863119896 minus119887] and 119875 =

(119860119896

119890119879

119899+1

)Compute 119862


120582120572119903(119862119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1


119896= diag119909119896 119860

119896= [119860119863

119896 minus119887] and 119875 = (

119860119896

119890119879

119899+1

)Compute 119862


120582120572119903(119862119901119862119901)

Then 119909119896+1 = 119879minus1119896

(119910119896+1) = 119863119896119910119896+1[119899]119910119870

119899+1

Let 119896 = 119896 + 1 and go to Step 3


119901119862119901) and 119909119896+1 = 119863

119896119910119896+1[119899]119910119870

119899+1





sum119894=1

ln (119909119894) (10)




1199100 minus 120582120572119903(119862119875119862119875) then

119892 (119910119896+1) minus 119892 (1199100) = nabla119892 (1199100)119879

(minus120582120572119903119862119875

10038171003817100381710038171198621198751003817100381710038171003817)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817(nabla119892 (1199100) 119862

119875)

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817

100381710038171003817100381711986211987510038171003817100381710038172

= minus1205821205721199031003817100381710038171003817119862119875

1003817100381710038171003817 lt 0

(11)



119892 (119910119896+1) le (1 minus120582120572

119899 + 1)119892 (1199100) (12)







119899 + 1)119892 (1199100) (13)



10038171003817100381710038171003817119910119896+1 minus 1199100

10038171003817100381710038171003817 =10038171003817100381710038171003817120579 (119910lowast minus 1199100)

10038171003817100381710038171003817 = 120582120572119903 997904rArr

10038171003817100381710038171003817(119910lowast minus 1199100)

10038171003817100381710038171003817 =120582120572119903

120579

(14)


100381710038171003817100381710038172

=1003817100381710038171003817119910lowast10038171003817100381710038172

+100381710038171003817100381710038171199100100381710038171003817100381710038172

minus 21199100119879

119910lowast (15)

and we have

1003817100381710038171003817119910lowast1003817100381710038171003817 =119899+1

sum119894=1

(119910lowast119894)2

le (119899+1

sum119894=1

119910119894

lowast)

2

= (119890119879119899+1

119910lowast)2

= 1

100381710038171003817100381710038171199100100381710038171003817100381710038172

=1003817100381710038171003817100381710038171003817

1

119899 + 1119890119899+1

1003817100381710038171003817100381710038171003817

2

=1

119899 + 1

1199100119879

119910lowast =1

119899 + 1119890119879119899+1

119910lowast =1

119899 + 1

(16)

Thus

10038171003817100381710038171003817119910lowast minus 1199100

100381710038171003817100381710038172

le 1 +1

119899 + 1minus

2

119899 + 1=

119899

119899 + 1le

119899 + 1

119899 + 2 (17)


(120582120572119903

120579)2

le119899 + 1

119899 + 2or

120582120572119903

120579le

radic119899 + 1

radic119899 + 2

(18)


119892 (119910119896+1) le 120579119892 (119910lowast) + (1 minus 120579) 119892 (1199100)

le (1 minus120582120572

119899 + 1)119892 (1199100)

(19)


1))119892(1199100)


Proof Consider


119891 (119909119896) minus 119911lowast)

minus119899

sum119894=1

ln(119909119896+1119894

119909119896119894

)

= (119899 + 1) ln(119892 (119910119896+1)

119892 (1199100))

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le (119899 + 1) ln(1 minus120582120572

119899 + 1)

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le minus120582120572 +12058221205722

2 (1 minus 120582120572)2

(20)


minus119899+1

sum119894=1

ln (119910119896+1119894

) le12058221205722

2 (1 minus 120582120572)2 (21)



(1) 1199090 ge 2minus119871119890119899



Proof Consider

119891 (119909119896) minus 119911lowast

119891 (1199090) minus 119911lowast= 120574 (119909119896) exp(

120601 (119909119896) minus 120601 (1199090)

119899 + 1) (22)

where

120574 (119909119896) = exp(sum119899

119894=1ln (119909119896119894) minus sum119899

119894=1ln (1199090119894)

119899 + 1) (23)


119891 (119909119896) minus 119911lowast


119899 + 1)

(24)




Example 8

1199091

11176285677 07993104755 07992885091 079846775721199092

08925922545 12006283419 12006478942 120137562451199093

minus00102208222 00000611826 00000635969 000015661831199094

13324440588 03980537916 03979927208 0395765081

Example 9

1199091

minus52199797446 07514629509 07514297188 075142749971199092

minus18871925195 00000032699 00000065967 000000681981199093

04463304312 00000010511 00000021234 000000220201199094

03791651531 10175937272 10172339995 101721015311199095

minus00087312913 00007335551 00007190583 000071809831199096

05536695688 09999989489 09999978766 099999779801199097

06232230772 00000006675 00000013472 00000013940

Example 101199091

00000000025 00000000156 00000000055 000000000801199092

19999999829 19999999858 19999999749 199999995861199093

00000000025 00000000156 00000000055 00000000080

Example 11

1199091

03124560511 07142857707 07142857350 071428574531199092

03438507936 01428572168 01428571684 014285716001199093

00001576431 00000002101 00000000702 000000004981199094

minus14064826279 00000001087 00000000363 00000000258



Example 8



Example 9



Example 10



Example 11




120601(1199090) le minus119896120575 Thus


119899 + 1)

le 25119871 exp(minus119896120575

119899 + 1) le 2minus4119871

(25)








100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8


0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11


5 Numerical Results




st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)








min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899 + 1)119892 (1199100) (13)



10038171003817100381710038171003817119910119896+1 minus 1199100

10038171003817100381710038171003817 =10038171003817100381710038171003817120579 (119910lowast minus 1199100)

10038171003817100381710038171003817 = 120582120572119903 997904rArr

10038171003817100381710038171003817(119910lowast minus 1199100)

10038171003817100381710038171003817 =120582120572119903

120579

(14)


100381710038171003817100381710038172

=1003817100381710038171003817119910lowast10038171003817100381710038172

+100381710038171003817100381710038171199100100381710038171003817100381710038172

minus 21199100119879

119910lowast (15)

and we have

1003817100381710038171003817119910lowast1003817100381710038171003817 =119899+1

sum119894=1

(119910lowast119894)2

le (119899+1

sum119894=1

119910119894

lowast)

2

= (119890119879119899+1

119910lowast)2

= 1

100381710038171003817100381710038171199100100381710038171003817100381710038172

=1003817100381710038171003817100381710038171003817

1

119899 + 1119890119899+1

1003817100381710038171003817100381710038171003817

2

=1

119899 + 1

1199100119879

119910lowast =1

119899 + 1119890119879119899+1

119910lowast =1

119899 + 1

(16)

Thus

10038171003817100381710038171003817119910lowast minus 1199100

100381710038171003817100381710038172

le 1 +1

119899 + 1minus

2

119899 + 1=

119899

119899 + 1le

119899 + 1

119899 + 2 (17)


(120582120572119903

120579)2

le119899 + 1

119899 + 2or

120582120572119903

120579le

radic119899 + 1

radic119899 + 2

(18)


119892 (119910119896+1) le 120579119892 (119910lowast) + (1 minus 120579) 119892 (1199100)

le (1 minus120582120572

119899 + 1)119892 (1199100)

(19)


1))119892(1199100)


Proof Consider


119891 (119909119896) minus 119911lowast)

minus119899

sum119894=1

ln(119909119896+1119894

119909119896119894

)

= (119899 + 1) ln(119892 (119910119896+1)

119892 (1199100))

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le (119899 + 1) ln(1 minus120582120572

119899 + 1)

minus119899+1

sum119894=1

ln (119910119896+1119894

)

le minus120582120572 +12058221205722

2 (1 minus 120582120572)2

(20)


minus119899+1

sum119894=1

ln (119910119896+1119894

) le12058221205722

2 (1 minus 120582120572)2 (21)



(1) 1199090 ge 2minus119871119890119899



Proof Consider

119891 (119909119896) minus 119911lowast

119891 (1199090) minus 119911lowast= 120574 (119909119896) exp(

120601 (119909119896) minus 120601 (1199090)

119899 + 1) (22)

where

120574 (119909119896) = exp(sum119899

119894=1ln (119909119896119894) minus sum119899

119894=1ln (1199090119894)

119899 + 1) (23)


119891 (119909119896) minus 119911lowast


119899 + 1)

(24)




Example 8

1199091

11176285677 07993104755 07992885091 079846775721199092

08925922545 12006283419 12006478942 120137562451199093

minus00102208222 00000611826 00000635969 000015661831199094

13324440588 03980537916 03979927208 0395765081

Example 9

1199091

minus52199797446 07514629509 07514297188 075142749971199092

minus18871925195 00000032699 00000065967 000000681981199093

04463304312 00000010511 00000021234 000000220201199094

03791651531 10175937272 10172339995 101721015311199095

minus00087312913 00007335551 00007190583 000071809831199096

05536695688 09999989489 09999978766 099999779801199097

06232230772 00000006675 00000013472 00000013940

Example 101199091

00000000025 00000000156 00000000055 000000000801199092

19999999829 19999999858 19999999749 199999995861199093

00000000025 00000000156 00000000055 00000000080

Example 11

1199091

03124560511 07142857707 07142857350 071428574531199092

03438507936 01428572168 01428571684 014285716001199093

00001576431 00000002101 00000000702 000000004981199094

minus14064826279 00000001087 00000000363 00000000258



Example 8



Example 9



Example 10



Example 11




120601(1199090) le minus119896120575 Thus


119899 + 1)

le 25119871 exp(minus119896120575

119899 + 1) le 2minus4119871

(25)








100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8


0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11


5 Numerical Results




st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)








min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Example 8

1199091

11176285677 07993104755 07992885091 079846775721199092

08925922545 12006283419 12006478942 120137562451199093

minus00102208222 00000611826 00000635969 000015661831199094

13324440588 03980537916 03979927208 0395765081

Example 9

1199091

minus52199797446 07514629509 07514297188 075142749971199092

minus18871925195 00000032699 00000065967 000000681981199093

04463304312 00000010511 00000021234 000000220201199094

03791651531 10175937272 10172339995 101721015311199095

minus00087312913 00007335551 00007190583 000071809831199096

05536695688 09999989489 09999978766 099999779801199097

06232230772 00000006675 00000013472 00000013940

Example 101199091

00000000025 00000000156 00000000055 000000000801199092

19999999829 19999999858 19999999749 199999995861199093

00000000025 00000000156 00000000055 00000000080

Example 11

1199091

03124560511 07142857707 07142857350 071428574531199092

03438507936 01428572168 01428571684 014285716001199093

00001576431 00000002101 00000000702 000000004981199094

minus14064826279 00000001087 00000000363 00000000258



Example 8



Example 9



Example 10



Example 11




120601(1199090) le minus119896120575 Thus


119899 + 1)

le 25119871 exp(minus119896120575

119899 + 1) le 2minus4119871

(25)








100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8


0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11


5 Numerical Results




st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)








min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











100

200

300

400

500

600

log(tolerance)

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKYMN-MKKY

(a) Example 8


0

20

40

60

80

100

120

140

160

180

200

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(b) Example 9


0

10

20

30

40

50

60

70

80

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(c) Example 10


0

10

20

30

40

50

60

70

Num

ber o

f ite

ratio

ns

MKKY

Sch-MKKY

MN-MKKY

(d) Example 11


5 Numerical Results




st 1199091+ 1199092+ 1199093= 2

minus 1199091+ 21199092+ 1199094= 2

1199091 119909

4ge 0

(26)








min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












min 119891 (119909) = 11990921+ 11990922minus 81199092+ 8

st 1199091+ 21199092+ 1199093= 4

1199091 1199092 1199093ge 0

(27)


min 119891 (119909) = 11990921+ 211990952minus 1199091

st 1199091+ 21199092minus 1199093= 1

minus 31199091+ 1199092+ 1199094= minus2

1199091 1199092 1199093 1199094ge 0

(28)

6 Conclusion





References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Novel Interior Point Algorithms for Solving Nonlinear Convex ...downloads.hindawi.com/journals/aor/2015/487271.pdf · 2019-07-31 · Research Article Novel Interior