Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 594693 6 pageshttpdxdoiorg1011552013594693

Research ArticleA Branch and Bound Reduced Algorithm for QuadraticProgramming Problems with Quadratic Constraints

Yuelin Gao Feifei Li and Siqiao Jin

Institute of Information amp System Science Beifang University of Nationalities Yinchuan 750021 China

Correspondence should be addressed to Yuelin Gao gaoyuelin263net

Received 15 July 2013 Accepted 19 September 2013

Academic Editor Dongdong Ge

Copyright copy 2013 Yuelin Gao 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 propose a branch and bound reduced algorithm for quadratic programming problems with quadratic constraints In thisalgorithm we determine the lower bound of the optimal value of original problem by constructing a linear relaxation programmingproblem At the same time in order to improve the degree of approximation and the convergence rate of acceleration a rectangularreduction strategy is used in the algorithm Numerical experiments show that the proposed algorithm is feasible and effective andcan solve small- and medium-sized problems

1 Introduction

Quadratic programming problemswith quadratic constraintsplay a very important role in global optimization becausequadratic functions are relatively simple functions among allnonlinear functions and quadratic functions can approachmany other functions Therefore it is necessary for usto research quadratic problems for researching nonlinearproblems better and quadratic programming problems withquadratic constraints have an important applications inScience and technology Then in spite of researching localoptimization problems or global optimization problemsquadratic programming problems have got extensive atten-tion it is obvious that researching this kind of problemsis very necessary In this paper we consider the followingquadratic programming problemswith quadratic constraints

min 1198910 (119909) = 1199091198791198760119909 + (1198890)119879

119909 + 1198880

st 119891119894 (119909) = 119909119879119876119894119909 + (119889119894)119879

119909 + 119888119894le 0

119894 = 1 2 119901119909 isin 119878 = 119909 isin 119877119899 119897 le 119909 le 119906

(119876119875)

where 119876119894 = (11990211989411989511198952

)119899times119899

are 119899-dimension symmetric matrices119889119894 = (119889119894

1 1198891198942 119889119894

119899)119879 isin 119877119899 119897 isin 119877119899 119906 isin 119877119899 119888

119894isin 119877 and 119894 =

0 1 119901

In recent years many researchers have researched thiskind of problems and made certain progress In [1] aneffective lower bound of the optimal value of original problemis provided using Lagrange lower estimate and the localoptimal solutions are obtained by Newton methods then toaccelerate the convergence of the global optimal solutionsthe local Newton methods are used A decompose-approachmethod is put forward in [2] Literature [3] organicallycombines the outer approximation method with the branchand bound technique and presents a new branch-reducealgorithm Literature [4] combines the cutting plane algo-rithm with the branch and bound algorithm and putsforwards a new algorithm Literature [5] presents a branchand bound algorithm by the linear lower function of thebilinear function Based on [5] literature [6] puts forwarda branch-reduce method aiming at objective function andconstraint conditions of the linear relaxation programmingA simplex branch and bound algorithm is raised in [7]There are many different methods for solving quadraticprogramming problems with quadratic constraints in [8ndash15]

The rest of this paper is organized as follows In Section 2we give the linear relaxation programming problem (119871119875)of the problem (119876119875) In Section 3 we give the rectanglesubdivision and reduce strategy We explain the branch andbound algorithm in detail in Section 4 and the convergence

2 Mathematical Problems in Engineering

of the algorithm is proved Finally some numerical resultsturn out the effectiveness of the present algorithm

2 Linear Relaxation Programming

In this section we construct a linear relaxation programmingproblem of the original problem

Assume that 120582119894min is the minimum eigenvalue of thematrices 119876119894 for 119894 = 1 2 119901 If 120582119894min ge 0 let 120579

119894= 0

otherwise let 120579119894= |120582119894min| + 120591119894 where 120591119894 ge 0 then 119876119894 + 120579

119894119868

is semipositive definiteOn the rectangle 119878119896 = 119909 isin 119877119899 119897119896 le 119909 le 119906119896 for each 119894

we construct a linear lower function119891119894(119909) on 119878119896We have

119891119894 (119909) = 119909119879119876119894119909 + (119889119894)

119879

119909 + 119888119894

= 119909119879 (119876119894 + 120579119894119868) 119909 + (119889119894)

119879

119909 + 119888119894minus 1205791198941199092

= (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + (119889119894)

119879

119909 + 119888119894minus 120579119894

119899

sum119895=1

1199092119895

+ 2(119897119896)119879

(119876119894 + 120579119894119868) 119909 minus (119897119896)

119879

(119876119894 + 120579119894119868) 119897119896

(1)

Suppose that 119897119896119895and 119906119896

119895are the 119895th indicators of 119897119896 and 119906119896

respectively We know that for each 119895 isin 1 2 119899 a linearlower function of minus1199092

119895is minus(119906119896

119895+ 119897119896119895)119909119895+ 119906119896119895119897119896119895on the interval

[119897119896119895 119906119896119895] Therefore

120593119878119896 (119909) ≜

119899

sum119895=1

(minus (119906119896119895+ 119897119896119895) 119909119895+ 119906119896119895119897119896119895)

= minus(119897119896 + 119906119896)119879

119909 + (119897119896)119879

119906119896

(2)

is a linear lower function ofminussum119899119895=1

1199092119895on the rectangle [119897119896 119906119896]

we construct the following linear function

119897119894119878119896 (119909) = (119886

119894

119878119896)119879

119909 + 119887119894119878119896 (3)

where119886119894119878119896 = 119889119894 + 2 (119876119894 + 120579

119894119868) 119897119896 minus 120579

119894(119897119896 + 119906119896)

119887119894119878119896 = 119888119894 minus (119897

119896)119879

(119876119894 + 120579119894119868) 119897119896 + 120579

119894(119897119896)119879

119906119896

(4)

We can obtain the following two theorems

Theorem 1 For each 119894 isin 0 1 119901 let 119876119894 + 120579119894119868 be

semipositive definite For each 119894 isin 0 1 119901 the linearfunction 119897119894

119878119896(119909) is a lower function of 119891119894(119909) on the rectangle 119878119896

that is 119891119894(119909) ge 119897119894119878119896(119909) for all 119909 isin 119878119896

Proof From the formula (1) and the definitions of thefunctions 120593

119878119896(119909) and 119897119894

119878119896(119909) for each 119894 isin 0 1 119901 we have

119891119894 (119909) ge (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + 119897119894

119878119896 (119909) forall119909 isin 119878119896

(5)

Moreover the matrix 119876119894 + 120579119894119868 is semipositive definite then

(119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) ge 0 forall119909 isin 119878119896 (6)

Consequently 119891119894(119909) ge 119897119894119878119896(119909) for all 119909 isin 119878119896 119894 isin 0 1 119901

Theorem 2 Assume that 120588(119876119894 + 120579119894119868) is the spectral radius of

the rectangle 119876119894 + 120579119894119868 then

max 10038161003816100381610038161003816119891119894(119909) minus 119897

119894

119878119896 (119909)

10038161003816100381610038161003816 119909 isin 119878119896

le (120588 (119876119894 + 120579119894119868) + 120579

119894)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

119894 isin 0 1 119901

(7)

Proof From the formula (1) and the definitions of thefunctions 120593

119878119896(119909) and 119897119894

119878119896(119909) we have

119891119894 (119909) minus 119897119894

119878119896 (119909)

= (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + 120579

119894(minus119909

2 minus 120593119878119896(119909))

le 120588 (119876119894 + 120579119894119868)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

+ 120579119894

1003816100381610038161003816100381610038161003816(119909 minus 119897119896)

119879

(119906119896 minus 119909)1003816100381610038161003816100381610038161003816

le 120588 (119876119894 + 120579119894119868)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

+ 120579119894

10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

= (120588 (119876119894 + 120579119894119868) + 120579

119894)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

(8)

Hence the conclusion is established

Therefore from Theorem 1 we obtain the linear relax-ation programming problem of (119876119875) on the rectangle 119878119896

min 1198970119878119896 (119909)

st 119897119894119878119896 (119909) le 0 119894 = 1 2 119901

119909 isin 119878119896

(119871119875 (119878119896))

Solving the problem (119871119875 (119878119896)) its optimal value is obtainedwhich is a lower bound of the global optimum of the problem(119876119875) on the rectangle 119878119896

3 The Subdivision andReduction of the Rectangle

In this section we give the bisection and reduction methodsof the rectangle Let 119878119896 = 119897119896 le 119909 le 119906119896 be a rectangle on 119877119899and 119909119896 isin 119878119896

31 The Subdivision of the Rectangle The method of thesubdivision of the rectangle is described as follows

(i) Select the longest edge of the rectangle 119878119896 that is119880119896119904minus

119871119896119904= max119880119896

119895minus 119871119896119895 119895 = 1 2 119899

Mathematical Problems in Engineering 3

(ii) Let 119881119896119904= (119880119896119904+ 119871119896119904)2 Then

1198781198961 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119871119896

119904 119881119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

1198781198962 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119881119896

119904 119880119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

(9)

32 The Reduction of the Rectangle Based on [8] in orderto improve the convergence of the algorithm we give twopruning methods of problem (119871119875) For all 119878119896 = 119909 isin 119877119899

119897119896 le 119909 le 119906119896 sube 119878 119878119896119895= [119897119896119895 119906119896119895] suppose that the objective

function of (119871119875 (119878119896)) is1205931198960(119909) = sum

119899

119895=1119888119896119895119909119895+1198881198960 the constraint

functions aresum119899119895=1

119886119896119894119895119909119895le 119887119896119894 and the upper bound of (119876119875) is

denoted by 119880119861 let

119903119862119896 =119899

sum119895=1

min 119888119896119895119897119896119895 119888119896119895119906119896119895 119903119871119896

119894=119899

sum119895=1

min 119886119896119894119895119897119896119895 119886119896119894119895119906119896119895

119894 = 1 119901

(10)

Theorem 3 (see [8]) For any 119878119896 sube 119878 if 119903119862119896 + 1198881198960gt 119880119861 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119888119896119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on119878119896

= (119878119896

119895)119899times1

if 119888119896119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119880119861minus119888119896

0minus119903119862119896+119888119896

119895119897119896119895

119888119896119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 =119903 119895=1 119899

[

[

119897119896119895119880119861minus119888119896

0minus119903119862119896+119888119896

119895119906119896119895

119888119896119895

) cap 119878119896119895 119895=119903

(11)

Theorem 4 (see [8]) For any 119894 = 1 119901 if 119903119871119896119894gt 119887119896119894 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119886119896119894119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on

119878119896

= (119878119896

119895)119899times1

if 119886119896119894119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119887119896119894minus119903119871119896119894+119886119896119894119895119897119896119895

119886119896119894119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 = 119903 119895=1 119899

[

[

119897119896119895119887119896119894minus119903119871119896119894+119886119896119894119895119906119896119895

119886119896119894119895

) cap 119878119896119895 119895=119903

(12)

From Theorems 3 and 4 we can construct the followingpruning rules to delete or reduce the rectangle 119878119896

Rule 1 Compute 119903119862119896 if 119903119862119896 + 1198881198960gt 119880119861 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119888119896119895gt 0 let 119906119896

119895= min119906119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119897119896119895)119888119896119895

If 119888119896119895lt 0 let 119897119896

119895= max119897119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119906119896119895)119888119896119895

Rule 2 Compute 119903119871119896119894 if 119903119871119896

119894gt 119887119896119894 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119886119896119894119895gt 0 let 119906119896

119895= min119906119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119897119896119895)119886119896119894119895

If 119886119896119894119895lt 0 let 119897119896

119895= max119897119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119906119896119895)119886119896119894119895 where

119894 = 1 119901

4 The Algorithm Description andConvergence Analysis

Next we can describe a branch and bound reduced algorithmof problem (119876119875) as follows

Supposewhen the iteration proceeds in step 119896 the feasibleregion of the problem (119876119875) is denoted by 119863 119876 representsthe feasible set at present 119878119896 represents the divided rectanglesoon the set of remained rectangle after pruning is denotedby 119879 and the current lower bound and upper bound of theglobal optimal value of the problem (119876119875) are denoted by 120572

119896

and 120573119896 respectively

Step 1 (initializing) Set 120576 gt 0 and let 119879 = 119878 119896 = 1 119878119896 = 119878and 120573

119896= infin Solving the problem (119871119875 (119878119896)) its optimal

solution and optimal value are denoted by 119909119896 and 120573(119878119896)respectively Let120573

119896= 120573(119878119896) then120573

119896is a lower bound of global

optimal value of the problem (119876119875) if119909119896 isin 119863 let119876 = 119876cup119909119896the upper bound is 119876 = 119876 cup 119909119896 and find a current optimalsolution 119909lowast isin argmin120572

119896

Step 2 (termination rule) If there was a condition satisfyingbetween 120572

119896minus 120573119896le 120576 (119896 = 1 2 ) or 119879 = 0 then stop

the global optimal solution 119909lowast and the global optimal value1198910(119909lowast) are outputted otherwise go to the next step

Step 3 (selection rule) Select a rectangle which has a min-imum lower bound in the rectangle set 119879 that is 119878119896 =argmin120573

119896

Step 4 (subdivision rule) Using the subdivision method inthe former section then the rectangle 119878119896 can be divided intosubrectangles 1198781198961 and 1198781198962 and int 1198781198961 cap int 1198781198962 = 0

Step 5 (reduction technique) Reducing the subrectanglesafter dividing using the reduction method in the former

4 Mathematical Problems in Engineering

section without loss of generality the new rectangles afterreduction are also denoted by 119878119896119895 119895 isin Γ where Γ is the indexset of the rectangles after reduction

Step 6 (bounding rule) Lower bound is 120573lowast119896= min120573

119896 119896 =

1 2 upper bound is 120572lowast119896= min1198910(119909) 119909 isin 119876

The current best feasible solution is 119909lowast isin argmin1198910(119909) 119909 isin 119876

Step 7 (pruning rule) Let 119879 = 119879 119878 120573119896(119878) ge 120572lowast

119896 119878 isin 119879

Step 8 Set 119896 = 119896 + 1 go to Step 2

Theorem 5 (a) If the algorithm terminates in limited stepsthen 119909119896 is a 120576-global optimal solution of problem (119876119875)

(b) For each 119896 ge 1 let 119909119896 be the solution after step 119896 If thealgorithm is infinite then 119909119896 is a feasible solution sequenceof problem (119876119875) and any accumulation is a global optimalsolution of problem (119876119875) and lim119896rarrinfin120572119896 = lim

119896rarrinfin120573119896= ]

Proof (a) If the algorithm is finite suppose that it terminatesin step 119896 (119896 ge 1) Because119909119896 is obtained by solving (119871119875 (119878119896))then 119909119896 isin 119878119896 sube 119878 and 119909119896 is a feasible solution of problem(119876119875) When 120572

119896minus120573119896le 120576 the algorithm terminate From Steps

1 and 6 we have 1198910(119909119896) minus 120573119896le 120576 from the algorithm 120573

119896le V

where V is the global optimal value of problem (119876119875) Because119909119896 is a feasible solution of problem (119876119875) so 1198910(119909119896) ge VThus

V le 1198910 (119909119896) le V + 120576 (13)

Therefore 119909119896 is a 120576-global optimal solution of problem(119876119875)

(b) If the algorithm is infinite then it produces a solutionsequence 119909119896 of problem (119876119875) where for each 119896 ge 1 119909119896 isobtained by solving problem (119871119875 (119878119896)) For each 119878119896 sube 119878 forthe optimal solution119909119896 isin 119878119896 sube 119878 the sequence 119909119896 constitutea solution sequence of problem (119876119875) from the iteration of thealgorithm we have

120573119896le V le 120572

119896= 1198910 (119909119896) 119896 = 1 2 (14)

Because the series 120573119896 do not decrease and have an upper

bound and 120572119896 do not increase and have a lower bound then

the series 120573119896 and 120572

119896 are both convergent Taking the limits

on both sides of (14) we have

lim119896rarrinfin

120573119896le V le lim

119896rarrinfin

120572119896= lim119896rarrinfin

1198910 (119909119896) (15)

Let lim119896rarrinfin

120573119896= 120573 lim

119896rarrinfin120572119896= 120572 then the formula (15)

converts into

120573 le V le lim119896rarrinfin

1198910 (119909119896) = 120572 (16)

Without loss of generality assume that the sequence ofrectangle 119878119896 = [119897119896 119906119896] satisfy 119909119896 isin 119878119896 and 119878119896+1 sub 119878119896

In our algorithm the rectangles are divided into two equalparts continuously then⋂infin

119896=1119878119896+1 = 119909119896 and because of the

continuity of function 1198910(119909)

120573 = V = 120572 = lim119896rarrinfin

1198910 (119909119896) = 1198910 (119909lowast) (17)

So any accumulation of 119909119896 is a global optimal solution ofproblem (119876119875)

5 Numerical Experiment

Several experiments are given to turn out the feasibility andeffectiveness of our algorithm

Example 1

min 11990921+ 11990922

st 0311990911199092ge 1

2 le 1199091le 5

1 le 1199092le 3

(18)

From the algorithm the initial rectangle is 1198781 =

[ 20000 5000010000 30000

] first we solve the problem 119871119875(1198781) its optimalsolution is 1199091 = (20000 30000) and optimal value is1205731= 120573(1198781) = 49996 then 49996 is a lower bound of

the global optimal value of problem (119876119875) Because 1199091 =(20000 30000) is feasible then 119876 = [20000 30000] is a setof current feasible solutions and the current upper boundis 1205721= 1198910(1199091) = 130000 the current optimal solution is

119909lowast = 1199091 = (20000 30000)After that based on our selection rule select the rectangle

with the minimum lower bound 1198781 to divide then 1198781 isdivided into two subrectangles 11987811 = [ 20000 35000

10000 30000] and

11987812 = [ 35000 5000010000 30000

] from the dividing method in Section 31then reduce the rectangles using the reduction technique inSection 32 and the new rectangle after reduction is denotedby 1198782 = 11987811 = [ 20000 35000

10000 30000] Solving the linear relaxation

programming problem 119871119875 on the rectangle 1198782 its optimalvalue is 120573

2= 120573(11987811) = 49996 then the lower bound

of the original problem is not updated also being 49996Next we choose 1198782 to divide until sdot sdot sdot the 15th iteration11987814 = [ 20000 20408

16538 17019] solve the linear relaxation programming

problem 119871119875(11987814) its optimal solution is (20000 16665) andoptimal value is 67765 while the current upper boundis 68151 the current optimal solution is (20000 16778)Because |68151 minus 67765| lt 01 it satisfies our terminationrule then the optimal value of the original problem is 68151the lower bound of the optimal value is 67765 and theoptimal solution is 119909 = (20000 16778) here the lowerbound of the optimal value is also approximate optimal valuewhere the accuracy is 120576 = 01

Table 2 shows the different results of Example 1 underdifferent accuracy

Mathematical Problems in Engineering 5

Table 1

Example The optimal solution within accuracy or one solution among solutions1199091

1199092

1199093

1199094

1199095

1 20000 166672 25576 312793 20000 166674 10000 550005 15000 122476 10156 155947 04267 058798 780000 330001 299958 449998 367753Example Approximate optimal value Iterations CPU (s)1 67778 33 83011282 1183837 49 326965653 67778 29 66764444 10000 9 31621065 minus11629 17 44298066 minus318878 130 540241407 minus33304 20 54109438 10128 98 193921992

Table 2 Different results of Example 1 under different accuracy

Example 1120576 Approximate optimal value Optimal value10119890 minus 2 6777772334392922 6784953802104409

10119890 minus 3 6777777695638590 6778685210977349

10119890 minus 4 6777777810403491 6777777840618791

Example 2

min 611990921+ 411990922+ 511990911199092

st minus611990911199092le minus48

0 le 1199091 1199092le 10

(19)

The optimal value is 1183838

Example 3

min 11990921+ 11990922

st minus0311990911199092le minus1

minus1199091minus 1199092le 1

119909 isin 1198830 = 2 le 1199091le 5 1 le 119909

2le 3

(20)

The optimal value is 67778

Example 4

min 1199091

st 1

41199091+1

21199092minus1

611990922minus1

611990921le 1

1

1411990921+1

1411990922minus3

71199091minus3

71199092le 1

1 le 1199091le 55 1 le 119909

2le 55

(21)

The optimal value is 10000

Example 5

min minus1199091+ 1199091119909052minus 1199092

st minus61199091+ 81199092le 3

31199091minus 1199092le 3

119909 isin 1198830 = 119909 | 0 le 119909119894le 15 119894 = 1 2

(22)

The optimal value is minus11629

Example 6

min 611990921+ 411990922+ 25(119909

1+ 1199092)2minus 25 (10119909

1+ 10119909

2)

st 3(1199091minus 1199092)2minus 3 (10119909

1+ 10119909

2) le minus48

0 le 1199091 1199092le 10

(23)

The optimal value is minus318878

Example 7

min 2111990921+ 34119909

11199092minus 241199092

2+ 21199091minus 14119909

2

st 211990921+ 411990911199092+ 211990922+ 81199091+ 61199092minus 9 le 0

minus511990921minus 811990911199092minus 511990922minus 41199091+ 41199092+ 4 le 0

1199091+ 21199092le 2 119909 isin [0 1]2

(24)

The optimal value is minus33205

6 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

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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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of the algorithm is proved Finally some numerical resultsturn out the effectiveness of the present algorithm

2 Linear Relaxation Programming

In this section we construct a linear relaxation programmingproblem of the original problem

Assume that 120582119894min is the minimum eigenvalue of thematrices 119876119894 for 119894 = 1 2 119901 If 120582119894min ge 0 let 120579

119894= 0

otherwise let 120579119894= |120582119894min| + 120591119894 where 120591119894 ge 0 then 119876119894 + 120579

119894119868

is semipositive definiteOn the rectangle 119878119896 = 119909 isin 119877119899 119897119896 le 119909 le 119906119896 for each 119894

we construct a linear lower function119891119894(119909) on 119878119896We have

119891119894 (119909) = 119909119879119876119894119909 + (119889119894)

119879

119909 + 119888119894

= 119909119879 (119876119894 + 120579119894119868) 119909 + (119889119894)

119879

119909 + 119888119894minus 1205791198941199092

= (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + (119889119894)

119879

119909 + 119888119894minus 120579119894

119899

sum119895=1

1199092119895

+ 2(119897119896)119879

(119876119894 + 120579119894119868) 119909 minus (119897119896)

119879

(119876119894 + 120579119894119868) 119897119896

(1)

Suppose that 119897119896119895and 119906119896

119895are the 119895th indicators of 119897119896 and 119906119896

respectively We know that for each 119895 isin 1 2 119899 a linearlower function of minus1199092

119895is minus(119906119896

119895+ 119897119896119895)119909119895+ 119906119896119895119897119896119895on the interval

[119897119896119895 119906119896119895] Therefore

120593119878119896 (119909) ≜

119899

sum119895=1

(minus (119906119896119895+ 119897119896119895) 119909119895+ 119906119896119895119897119896119895)

= minus(119897119896 + 119906119896)119879

119909 + (119897119896)119879

119906119896

(2)

is a linear lower function ofminussum119899119895=1

1199092119895on the rectangle [119897119896 119906119896]

we construct the following linear function

119897119894119878119896 (119909) = (119886

119894

119878119896)119879

119909 + 119887119894119878119896 (3)

where119886119894119878119896 = 119889119894 + 2 (119876119894 + 120579

119894119868) 119897119896 minus 120579

119894(119897119896 + 119906119896)

119887119894119878119896 = 119888119894 minus (119897

119896)119879

(119876119894 + 120579119894119868) 119897119896 + 120579

119894(119897119896)119879

119906119896

(4)

We can obtain the following two theorems

Theorem 1 For each 119894 isin 0 1 119901 let 119876119894 + 120579119894119868 be

semipositive definite For each 119894 isin 0 1 119901 the linearfunction 119897119894

119878119896(119909) is a lower function of 119891119894(119909) on the rectangle 119878119896

that is 119891119894(119909) ge 119897119894119878119896(119909) for all 119909 isin 119878119896

Proof From the formula (1) and the definitions of thefunctions 120593

119878119896(119909) and 119897119894

119878119896(119909) for each 119894 isin 0 1 119901 we have

119891119894 (119909) ge (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + 119897119894

119878119896 (119909) forall119909 isin 119878119896

(5)

Moreover the matrix 119876119894 + 120579119894119868 is semipositive definite then

(119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) ge 0 forall119909 isin 119878119896 (6)

Consequently 119891119894(119909) ge 119897119894119878119896(119909) for all 119909 isin 119878119896 119894 isin 0 1 119901

Theorem 2 Assume that 120588(119876119894 + 120579119894119868) is the spectral radius of

the rectangle 119876119894 + 120579119894119868 then

max 10038161003816100381610038161003816119891119894(119909) minus 119897

119894

119878119896 (119909)

10038161003816100381610038161003816 119909 isin 119878119896

le (120588 (119876119894 + 120579119894119868) + 120579

119894)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

119894 isin 0 1 119901

(7)

Proof From the formula (1) and the definitions of thefunctions 120593

119878119896(119909) and 119897119894

119878119896(119909) we have

119891119894 (119909) minus 119897119894

119878119896 (119909)

= (119909 minus 119897119896)119879

(119876119894 + 120579119894119868) (119909 minus 119897119896) + 120579

119894(minus119909

2 minus 120593119878119896(119909))

le 120588 (119876119894 + 120579119894119868)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

+ 120579119894

1003816100381610038161003816100381610038161003816(119909 minus 119897119896)

119879

(119906119896 minus 119909)1003816100381610038161003816100381610038161003816

le 120588 (119876119894 + 120579119894119868)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

+ 120579119894

10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

= (120588 (119876119894 + 120579119894119868) + 120579

119894)10038171003817100381710038171003817119906119896 minus 119897119896

100381710038171003817100381710038172

(8)

Hence the conclusion is established

Therefore from Theorem 1 we obtain the linear relax-ation programming problem of (119876119875) on the rectangle 119878119896

min 1198970119878119896 (119909)

st 119897119894119878119896 (119909) le 0 119894 = 1 2 119901

119909 isin 119878119896

(119871119875 (119878119896))

Solving the problem (119871119875 (119878119896)) its optimal value is obtainedwhich is a lower bound of the global optimum of the problem(119876119875) on the rectangle 119878119896

3 The Subdivision andReduction of the Rectangle

In this section we give the bisection and reduction methodsof the rectangle Let 119878119896 = 119897119896 le 119909 le 119906119896 be a rectangle on 119877119899and 119909119896 isin 119878119896

31 The Subdivision of the Rectangle The method of thesubdivision of the rectangle is described as follows

(i) Select the longest edge of the rectangle 119878119896 that is119880119896119904minus

119871119896119904= max119880119896

119895minus 119871119896119895 119895 = 1 2 119899

Mathematical Problems in Engineering 3

(ii) Let 119881119896119904= (119880119896119904+ 119871119896119904)2 Then

1198781198961 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119871119896

119904 119881119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

1198781198962 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119881119896

119904 119880119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

(9)

32 The Reduction of the Rectangle Based on [8] in orderto improve the convergence of the algorithm we give twopruning methods of problem (119871119875) For all 119878119896 = 119909 isin 119877119899

119897119896 le 119909 le 119906119896 sube 119878 119878119896119895= [119897119896119895 119906119896119895] suppose that the objective

function of (119871119875 (119878119896)) is1205931198960(119909) = sum

119899

119895=1119888119896119895119909119895+1198881198960 the constraint

functions aresum119899119895=1

119886119896119894119895119909119895le 119887119896119894 and the upper bound of (119876119875) is

denoted by 119880119861 let

119903119862119896 =119899

sum119895=1

min 119888119896119895119897119896119895 119888119896119895119906119896119895 119903119871119896

119894=119899

sum119895=1

min 119886119896119894119895119897119896119895 119886119896119894119895119906119896119895

119894 = 1 119901

(10)

Theorem 3 (see [8]) For any 119878119896 sube 119878 if 119903119862119896 + 1198881198960gt 119880119861 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119888119896119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on119878119896

= (119878119896

119895)119899times1

if 119888119896119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119880119861minus119888119896

0minus119903119862119896+119888119896

119895119897119896119895

119888119896119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 =119903 119895=1 119899

[

[

119897119896119895119880119861minus119888119896

0minus119903119862119896+119888119896

119895119906119896119895

119888119896119895

) cap 119878119896119895 119895=119903

(11)

Theorem 4 (see [8]) For any 119894 = 1 119901 if 119903119871119896119894gt 119887119896119894 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119886119896119894119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on

119878119896

= (119878119896

119895)119899times1

if 119886119896119894119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119887119896119894minus119903119871119896119894+119886119896119894119895119897119896119895

119886119896119894119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 = 119903 119895=1 119899

[

[

119897119896119895119887119896119894minus119903119871119896119894+119886119896119894119895119906119896119895

119886119896119894119895

) cap 119878119896119895 119895=119903

(12)

From Theorems 3 and 4 we can construct the followingpruning rules to delete or reduce the rectangle 119878119896

Rule 1 Compute 119903119862119896 if 119903119862119896 + 1198881198960gt 119880119861 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119888119896119895gt 0 let 119906119896

119895= min119906119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119897119896119895)119888119896119895

If 119888119896119895lt 0 let 119897119896

119895= max119897119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119906119896119895)119888119896119895

Rule 2 Compute 119903119871119896119894 if 119903119871119896

119894gt 119887119896119894 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119886119896119894119895gt 0 let 119906119896

119895= min119906119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119897119896119895)119886119896119894119895

If 119886119896119894119895lt 0 let 119897119896

119895= max119897119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119906119896119895)119886119896119894119895 where

119894 = 1 119901

4 The Algorithm Description andConvergence Analysis

Next we can describe a branch and bound reduced algorithmof problem (119876119875) as follows

Supposewhen the iteration proceeds in step 119896 the feasibleregion of the problem (119876119875) is denoted by 119863 119876 representsthe feasible set at present 119878119896 represents the divided rectanglesoon the set of remained rectangle after pruning is denotedby 119879 and the current lower bound and upper bound of theglobal optimal value of the problem (119876119875) are denoted by 120572

119896

and 120573119896 respectively

Step 1 (initializing) Set 120576 gt 0 and let 119879 = 119878 119896 = 1 119878119896 = 119878and 120573

119896= infin Solving the problem (119871119875 (119878119896)) its optimal

solution and optimal value are denoted by 119909119896 and 120573(119878119896)respectively Let120573

119896= 120573(119878119896) then120573

119896is a lower bound of global

optimal value of the problem (119876119875) if119909119896 isin 119863 let119876 = 119876cup119909119896the upper bound is 119876 = 119876 cup 119909119896 and find a current optimalsolution 119909lowast isin argmin120572

119896

Step 2 (termination rule) If there was a condition satisfyingbetween 120572

119896minus 120573119896le 120576 (119896 = 1 2 ) or 119879 = 0 then stop

the global optimal solution 119909lowast and the global optimal value1198910(119909lowast) are outputted otherwise go to the next step

Step 3 (selection rule) Select a rectangle which has a min-imum lower bound in the rectangle set 119879 that is 119878119896 =argmin120573

119896

Step 4 (subdivision rule) Using the subdivision method inthe former section then the rectangle 119878119896 can be divided intosubrectangles 1198781198961 and 1198781198962 and int 1198781198961 cap int 1198781198962 = 0

Step 5 (reduction technique) Reducing the subrectanglesafter dividing using the reduction method in the former

4 Mathematical Problems in Engineering

section without loss of generality the new rectangles afterreduction are also denoted by 119878119896119895 119895 isin Γ where Γ is the indexset of the rectangles after reduction

Step 6 (bounding rule) Lower bound is 120573lowast119896= min120573

119896 119896 =

1 2 upper bound is 120572lowast119896= min1198910(119909) 119909 isin 119876

The current best feasible solution is 119909lowast isin argmin1198910(119909) 119909 isin 119876

Step 7 (pruning rule) Let 119879 = 119879 119878 120573119896(119878) ge 120572lowast

119896 119878 isin 119879

Step 8 Set 119896 = 119896 + 1 go to Step 2

Theorem 5 (a) If the algorithm terminates in limited stepsthen 119909119896 is a 120576-global optimal solution of problem (119876119875)

(b) For each 119896 ge 1 let 119909119896 be the solution after step 119896 If thealgorithm is infinite then 119909119896 is a feasible solution sequenceof problem (119876119875) and any accumulation is a global optimalsolution of problem (119876119875) and lim119896rarrinfin120572119896 = lim

119896rarrinfin120573119896= ]

Proof (a) If the algorithm is finite suppose that it terminatesin step 119896 (119896 ge 1) Because119909119896 is obtained by solving (119871119875 (119878119896))then 119909119896 isin 119878119896 sube 119878 and 119909119896 is a feasible solution of problem(119876119875) When 120572

119896minus120573119896le 120576 the algorithm terminate From Steps

1 and 6 we have 1198910(119909119896) minus 120573119896le 120576 from the algorithm 120573

119896le V

where V is the global optimal value of problem (119876119875) Because119909119896 is a feasible solution of problem (119876119875) so 1198910(119909119896) ge VThus

V le 1198910 (119909119896) le V + 120576 (13)

Therefore 119909119896 is a 120576-global optimal solution of problem(119876119875)

(b) If the algorithm is infinite then it produces a solutionsequence 119909119896 of problem (119876119875) where for each 119896 ge 1 119909119896 isobtained by solving problem (119871119875 (119878119896)) For each 119878119896 sube 119878 forthe optimal solution119909119896 isin 119878119896 sube 119878 the sequence 119909119896 constitutea solution sequence of problem (119876119875) from the iteration of thealgorithm we have

120573119896le V le 120572

119896= 1198910 (119909119896) 119896 = 1 2 (14)

Because the series 120573119896 do not decrease and have an upper

bound and 120572119896 do not increase and have a lower bound then

the series 120573119896 and 120572

119896 are both convergent Taking the limits

on both sides of (14) we have

lim119896rarrinfin

120573119896le V le lim

119896rarrinfin

120572119896= lim119896rarrinfin

1198910 (119909119896) (15)

Let lim119896rarrinfin

120573119896= 120573 lim

119896rarrinfin120572119896= 120572 then the formula (15)

converts into

120573 le V le lim119896rarrinfin

1198910 (119909119896) = 120572 (16)

Without loss of generality assume that the sequence ofrectangle 119878119896 = [119897119896 119906119896] satisfy 119909119896 isin 119878119896 and 119878119896+1 sub 119878119896

In our algorithm the rectangles are divided into two equalparts continuously then⋂infin

119896=1119878119896+1 = 119909119896 and because of the

continuity of function 1198910(119909)

120573 = V = 120572 = lim119896rarrinfin

1198910 (119909119896) = 1198910 (119909lowast) (17)

So any accumulation of 119909119896 is a global optimal solution ofproblem (119876119875)

5 Numerical Experiment

Several experiments are given to turn out the feasibility andeffectiveness of our algorithm

Example 1

min 11990921+ 11990922

st 0311990911199092ge 1

2 le 1199091le 5

1 le 1199092le 3

(18)

From the algorithm the initial rectangle is 1198781 =

[ 20000 5000010000 30000

] first we solve the problem 119871119875(1198781) its optimalsolution is 1199091 = (20000 30000) and optimal value is1205731= 120573(1198781) = 49996 then 49996 is a lower bound of

the global optimal value of problem (119876119875) Because 1199091 =(20000 30000) is feasible then 119876 = [20000 30000] is a setof current feasible solutions and the current upper boundis 1205721= 1198910(1199091) = 130000 the current optimal solution is

119909lowast = 1199091 = (20000 30000)After that based on our selection rule select the rectangle

with the minimum lower bound 1198781 to divide then 1198781 isdivided into two subrectangles 11987811 = [ 20000 35000

10000 30000] and

11987812 = [ 35000 5000010000 30000

] from the dividing method in Section 31then reduce the rectangles using the reduction technique inSection 32 and the new rectangle after reduction is denotedby 1198782 = 11987811 = [ 20000 35000

10000 30000] Solving the linear relaxation

programming problem 119871119875 on the rectangle 1198782 its optimalvalue is 120573

2= 120573(11987811) = 49996 then the lower bound

of the original problem is not updated also being 49996Next we choose 1198782 to divide until sdot sdot sdot the 15th iteration11987814 = [ 20000 20408

16538 17019] solve the linear relaxation programming

problem 119871119875(11987814) its optimal solution is (20000 16665) andoptimal value is 67765 while the current upper boundis 68151 the current optimal solution is (20000 16778)Because |68151 minus 67765| lt 01 it satisfies our terminationrule then the optimal value of the original problem is 68151the lower bound of the optimal value is 67765 and theoptimal solution is 119909 = (20000 16778) here the lowerbound of the optimal value is also approximate optimal valuewhere the accuracy is 120576 = 01

Table 2 shows the different results of Example 1 underdifferent accuracy

Mathematical Problems in Engineering 5

Table 1

Example The optimal solution within accuracy or one solution among solutions1199091

1199092

1199093

1199094

1199095

1 20000 166672 25576 312793 20000 166674 10000 550005 15000 122476 10156 155947 04267 058798 780000 330001 299958 449998 367753Example Approximate optimal value Iterations CPU (s)1 67778 33 83011282 1183837 49 326965653 67778 29 66764444 10000 9 31621065 minus11629 17 44298066 minus318878 130 540241407 minus33304 20 54109438 10128 98 193921992

Table 2 Different results of Example 1 under different accuracy

Example 1120576 Approximate optimal value Optimal value10119890 minus 2 6777772334392922 6784953802104409

10119890 minus 3 6777777695638590 6778685210977349

10119890 minus 4 6777777810403491 6777777840618791

Example 2

min 611990921+ 411990922+ 511990911199092

st minus611990911199092le minus48

0 le 1199091 1199092le 10

(19)

The optimal value is 1183838

Example 3

min 11990921+ 11990922

st minus0311990911199092le minus1

minus1199091minus 1199092le 1

119909 isin 1198830 = 2 le 1199091le 5 1 le 119909

2le 3

(20)

The optimal value is 67778

Example 4

min 1199091

st 1

41199091+1

21199092minus1

611990922minus1

611990921le 1

1

1411990921+1

1411990922minus3

71199091minus3

71199092le 1

1 le 1199091le 55 1 le 119909

2le 55

(21)

The optimal value is 10000

Example 5

min minus1199091+ 1199091119909052minus 1199092

st minus61199091+ 81199092le 3

31199091minus 1199092le 3

119909 isin 1198830 = 119909 | 0 le 119909119894le 15 119894 = 1 2

(22)

The optimal value is minus11629

Example 6

min 611990921+ 411990922+ 25(119909

1+ 1199092)2minus 25 (10119909

1+ 10119909

2)

st 3(1199091minus 1199092)2minus 3 (10119909

1+ 10119909

2) le minus48

0 le 1199091 1199092le 10

(23)

The optimal value is minus318878

Example 7

min 2111990921+ 34119909

11199092minus 241199092

2+ 21199091minus 14119909

2

st 211990921+ 411990911199092+ 211990922+ 81199091+ 61199092minus 9 le 0

minus511990921minus 811990911199092minus 511990922minus 41199091+ 41199092+ 4 le 0

1199091+ 21199092le 2 119909 isin [0 1]2

(24)

The optimal value is minus33205

6 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

(ii) Let 119881119896119904= (119880119896119904+ 119871119896119904)2 Then

1198781198961 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119871119896

119904 119881119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

1198781198962 =119904minus1

prod119895=1

[119871119896119895 119880119896119895] times [119881119896

119904 119880119896119904] times119899

prod119895=119904+1

[119871119896119895 119880119896119895]

(9)

32 The Reduction of the Rectangle Based on [8] in orderto improve the convergence of the algorithm we give twopruning methods of problem (119871119875) For all 119878119896 = 119909 isin 119877119899

119897119896 le 119909 le 119906119896 sube 119878 119878119896119895= [119897119896119895 119906119896119895] suppose that the objective

function of (119871119875 (119878119896)) is1205931198960(119909) = sum

119899

119895=1119888119896119895119909119895+1198881198960 the constraint

functions aresum119899119895=1

119886119896119894119895119909119895le 119887119896119894 and the upper bound of (119876119875) is

denoted by 119880119861 let

119903119862119896 =119899

sum119895=1

min 119888119896119895119897119896119895 119888119896119895119906119896119895 119903119871119896

119894=119899

sum119895=1

min 119886119896119894119895119897119896119895 119886119896119894119895119906119896119895

119894 = 1 119901

(10)

Theorem 3 (see [8]) For any 119878119896 sube 119878 if 119903119862119896 + 1198881198960gt 119880119861 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119888119896119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on119878119896

= (119878119896

119895)119899times1

if 119888119896119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119880119861minus119888119896

0minus119903119862119896+119888119896

119895119897119896119895

119888119896119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 =119903 119895=1 119899

[

[

119897119896119895119880119861minus119888119896

0minus119903119862119896+119888119896

119895119906119896119895

119888119896119895

) cap 119878119896119895 119895=119903

(11)

Theorem 4 (see [8]) For any 119894 = 1 119901 if 119903119871119896119894gt 119887119896119894 then

there is no optimal solution of (119876119875) on 119878119896 otherwise if 119886119896119894119903gt

0 (119903 isin 1 119899) then there is no optimal solution of (119876119875) on

119878119896

= (119878119896

119895)119899times1

if 119886119896119894119903lt 0 (119903 isin 1 119899) then there is no optimal

solution of (119876119875) on 119878119896 = (119878119896119895)119899times1

where

119878119896

119895=

119878119896119895 119895 =119903 119895=1 119899

(119887119896119894minus119903119871119896119894+119886119896119894119895119897119896119895

119886119896119894119895

119906119896119895]

]

cap 119878119896119895 119895=119903

119878119896119895=

119878119896119895 119895 = 119903 119895=1 119899

[

[

119897119896119895119887119896119894minus119903119871119896119894+119886119896119894119895119906119896119895

119886119896119894119895

) cap 119878119896119895 119895=119903

(12)

From Theorems 3 and 4 we can construct the followingpruning rules to delete or reduce the rectangle 119878119896

Rule 1 Compute 119903119862119896 if 119903119862119896 + 1198881198960gt 119880119861 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119888119896119895gt 0 let 119906119896

119895= min119906119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119897119896119895)119888119896119895

If 119888119896119895lt 0 let 119897119896

119895= max119897119896

119895 (119880119861 minus 119888119896

0minus 119903119862119896 + 119888119896

119895119906119896119895)119888119896119895

Rule 2 Compute 119903119871119896119894 if 119903119871119896

119894gt 119887119896119894 then 119878119896 is deleted

otherwise for any 119895 = 1 119899If 119886119896119894119895gt 0 let 119906119896

119895= min119906119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119897119896119895)119886119896119894119895

If 119886119896119894119895lt 0 let 119897119896

119895= max119897119896

119895 (119887119896119894minus 119903119871119896119894+ 119886119896119894119895119906119896119895)119886119896119894119895 where

119894 = 1 119901

4 The Algorithm Description andConvergence Analysis

Next we can describe a branch and bound reduced algorithmof problem (119876119875) as follows

Supposewhen the iteration proceeds in step 119896 the feasibleregion of the problem (119876119875) is denoted by 119863 119876 representsthe feasible set at present 119878119896 represents the divided rectanglesoon the set of remained rectangle after pruning is denotedby 119879 and the current lower bound and upper bound of theglobal optimal value of the problem (119876119875) are denoted by 120572

119896

and 120573119896 respectively

Step 1 (initializing) Set 120576 gt 0 and let 119879 = 119878 119896 = 1 119878119896 = 119878and 120573

119896= infin Solving the problem (119871119875 (119878119896)) its optimal

solution and optimal value are denoted by 119909119896 and 120573(119878119896)respectively Let120573

119896= 120573(119878119896) then120573

119896is a lower bound of global

optimal value of the problem (119876119875) if119909119896 isin 119863 let119876 = 119876cup119909119896the upper bound is 119876 = 119876 cup 119909119896 and find a current optimalsolution 119909lowast isin argmin120572

119896

Step 2 (termination rule) If there was a condition satisfyingbetween 120572

119896minus 120573119896le 120576 (119896 = 1 2 ) or 119879 = 0 then stop

the global optimal solution 119909lowast and the global optimal value1198910(119909lowast) are outputted otherwise go to the next step

Step 3 (selection rule) Select a rectangle which has a min-imum lower bound in the rectangle set 119879 that is 119878119896 =argmin120573

119896

Step 4 (subdivision rule) Using the subdivision method inthe former section then the rectangle 119878119896 can be divided intosubrectangles 1198781198961 and 1198781198962 and int 1198781198961 cap int 1198781198962 = 0

Step 5 (reduction technique) Reducing the subrectanglesafter dividing using the reduction method in the former

4 Mathematical Problems in Engineering

section without loss of generality the new rectangles afterreduction are also denoted by 119878119896119895 119895 isin Γ where Γ is the indexset of the rectangles after reduction

Step 6 (bounding rule) Lower bound is 120573lowast119896= min120573

119896 119896 =

1 2 upper bound is 120572lowast119896= min1198910(119909) 119909 isin 119876

The current best feasible solution is 119909lowast isin argmin1198910(119909) 119909 isin 119876

Step 7 (pruning rule) Let 119879 = 119879 119878 120573119896(119878) ge 120572lowast

119896 119878 isin 119879

Step 8 Set 119896 = 119896 + 1 go to Step 2

Theorem 5 (a) If the algorithm terminates in limited stepsthen 119909119896 is a 120576-global optimal solution of problem (119876119875)

(b) For each 119896 ge 1 let 119909119896 be the solution after step 119896 If thealgorithm is infinite then 119909119896 is a feasible solution sequenceof problem (119876119875) and any accumulation is a global optimalsolution of problem (119876119875) and lim119896rarrinfin120572119896 = lim

119896rarrinfin120573119896= ]

Proof (a) If the algorithm is finite suppose that it terminatesin step 119896 (119896 ge 1) Because119909119896 is obtained by solving (119871119875 (119878119896))then 119909119896 isin 119878119896 sube 119878 and 119909119896 is a feasible solution of problem(119876119875) When 120572

119896minus120573119896le 120576 the algorithm terminate From Steps

1 and 6 we have 1198910(119909119896) minus 120573119896le 120576 from the algorithm 120573

119896le V

where V is the global optimal value of problem (119876119875) Because119909119896 is a feasible solution of problem (119876119875) so 1198910(119909119896) ge VThus

V le 1198910 (119909119896) le V + 120576 (13)

Therefore 119909119896 is a 120576-global optimal solution of problem(119876119875)

(b) If the algorithm is infinite then it produces a solutionsequence 119909119896 of problem (119876119875) where for each 119896 ge 1 119909119896 isobtained by solving problem (119871119875 (119878119896)) For each 119878119896 sube 119878 forthe optimal solution119909119896 isin 119878119896 sube 119878 the sequence 119909119896 constitutea solution sequence of problem (119876119875) from the iteration of thealgorithm we have

120573119896le V le 120572

119896= 1198910 (119909119896) 119896 = 1 2 (14)

Because the series 120573119896 do not decrease and have an upper

bound and 120572119896 do not increase and have a lower bound then

the series 120573119896 and 120572

119896 are both convergent Taking the limits

on both sides of (14) we have

lim119896rarrinfin

120573119896le V le lim

119896rarrinfin

120572119896= lim119896rarrinfin

1198910 (119909119896) (15)

Let lim119896rarrinfin

120573119896= 120573 lim

119896rarrinfin120572119896= 120572 then the formula (15)

converts into

120573 le V le lim119896rarrinfin

1198910 (119909119896) = 120572 (16)

Without loss of generality assume that the sequence ofrectangle 119878119896 = [119897119896 119906119896] satisfy 119909119896 isin 119878119896 and 119878119896+1 sub 119878119896

In our algorithm the rectangles are divided into two equalparts continuously then⋂infin

119896=1119878119896+1 = 119909119896 and because of the

continuity of function 1198910(119909)

120573 = V = 120572 = lim119896rarrinfin

1198910 (119909119896) = 1198910 (119909lowast) (17)

So any accumulation of 119909119896 is a global optimal solution ofproblem (119876119875)

5 Numerical Experiment

Several experiments are given to turn out the feasibility andeffectiveness of our algorithm

Example 1

min 11990921+ 11990922

st 0311990911199092ge 1

2 le 1199091le 5

1 le 1199092le 3

(18)

From the algorithm the initial rectangle is 1198781 =

[ 20000 5000010000 30000

] first we solve the problem 119871119875(1198781) its optimalsolution is 1199091 = (20000 30000) and optimal value is1205731= 120573(1198781) = 49996 then 49996 is a lower bound of

the global optimal value of problem (119876119875) Because 1199091 =(20000 30000) is feasible then 119876 = [20000 30000] is a setof current feasible solutions and the current upper boundis 1205721= 1198910(1199091) = 130000 the current optimal solution is

119909lowast = 1199091 = (20000 30000)After that based on our selection rule select the rectangle

with the minimum lower bound 1198781 to divide then 1198781 isdivided into two subrectangles 11987811 = [ 20000 35000

10000 30000] and

11987812 = [ 35000 5000010000 30000

] from the dividing method in Section 31then reduce the rectangles using the reduction technique inSection 32 and the new rectangle after reduction is denotedby 1198782 = 11987811 = [ 20000 35000

10000 30000] Solving the linear relaxation

programming problem 119871119875 on the rectangle 1198782 its optimalvalue is 120573

2= 120573(11987811) = 49996 then the lower bound

of the original problem is not updated also being 49996Next we choose 1198782 to divide until sdot sdot sdot the 15th iteration11987814 = [ 20000 20408

16538 17019] solve the linear relaxation programming

problem 119871119875(11987814) its optimal solution is (20000 16665) andoptimal value is 67765 while the current upper boundis 68151 the current optimal solution is (20000 16778)Because |68151 minus 67765| lt 01 it satisfies our terminationrule then the optimal value of the original problem is 68151the lower bound of the optimal value is 67765 and theoptimal solution is 119909 = (20000 16778) here the lowerbound of the optimal value is also approximate optimal valuewhere the accuracy is 120576 = 01

Table 2 shows the different results of Example 1 underdifferent accuracy

Mathematical Problems in Engineering 5

Table 1

Example The optimal solution within accuracy or one solution among solutions1199091

1199092

1199093

1199094

1199095

1 20000 166672 25576 312793 20000 166674 10000 550005 15000 122476 10156 155947 04267 058798 780000 330001 299958 449998 367753Example Approximate optimal value Iterations CPU (s)1 67778 33 83011282 1183837 49 326965653 67778 29 66764444 10000 9 31621065 minus11629 17 44298066 minus318878 130 540241407 minus33304 20 54109438 10128 98 193921992

Table 2 Different results of Example 1 under different accuracy

Example 1120576 Approximate optimal value Optimal value10119890 minus 2 6777772334392922 6784953802104409

10119890 minus 3 6777777695638590 6778685210977349

10119890 minus 4 6777777810403491 6777777840618791

Example 2

min 611990921+ 411990922+ 511990911199092

st minus611990911199092le minus48

0 le 1199091 1199092le 10

(19)

The optimal value is 1183838

Example 3

min 11990921+ 11990922

st minus0311990911199092le minus1

minus1199091minus 1199092le 1

119909 isin 1198830 = 2 le 1199091le 5 1 le 119909

2le 3

(20)

The optimal value is 67778

Example 4

min 1199091

st 1

41199091+1

21199092minus1

611990922minus1

611990921le 1

1

1411990921+1

1411990922minus3

71199091minus3

71199092le 1

1 le 1199091le 55 1 le 119909

2le 55

(21)

The optimal value is 10000

Example 5

min minus1199091+ 1199091119909052minus 1199092

st minus61199091+ 81199092le 3

31199091minus 1199092le 3

119909 isin 1198830 = 119909 | 0 le 119909119894le 15 119894 = 1 2

(22)

The optimal value is minus11629

Example 6

min 611990921+ 411990922+ 25(119909

1+ 1199092)2minus 25 (10119909

1+ 10119909

2)

st 3(1199091minus 1199092)2minus 3 (10119909

1+ 10119909

2) le minus48

0 le 1199091 1199092le 10

(23)

The optimal value is minus318878

Example 7

min 2111990921+ 34119909

11199092minus 241199092

2+ 21199091minus 14119909

2

st 211990921+ 411990911199092+ 211990922+ 81199091+ 61199092minus 9 le 0

minus511990921minus 811990911199092minus 511990922minus 41199091+ 41199092+ 4 le 0

1199091+ 21199092le 2 119909 isin [0 1]2

(24)

The optimal value is minus33205

6 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

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 Mathematical Problems in Engineering

section without loss of generality the new rectangles afterreduction are also denoted by 119878119896119895 119895 isin Γ where Γ is the indexset of the rectangles after reduction

Step 6 (bounding rule) Lower bound is 120573lowast119896= min120573

119896 119896 =

1 2 upper bound is 120572lowast119896= min1198910(119909) 119909 isin 119876

The current best feasible solution is 119909lowast isin argmin1198910(119909) 119909 isin 119876

Step 7 (pruning rule) Let 119879 = 119879 119878 120573119896(119878) ge 120572lowast

119896 119878 isin 119879

Step 8 Set 119896 = 119896 + 1 go to Step 2

Theorem 5 (a) If the algorithm terminates in limited stepsthen 119909119896 is a 120576-global optimal solution of problem (119876119875)

(b) For each 119896 ge 1 let 119909119896 be the solution after step 119896 If thealgorithm is infinite then 119909119896 is a feasible solution sequenceof problem (119876119875) and any accumulation is a global optimalsolution of problem (119876119875) and lim119896rarrinfin120572119896 = lim

119896rarrinfin120573119896= ]

Proof (a) If the algorithm is finite suppose that it terminatesin step 119896 (119896 ge 1) Because119909119896 is obtained by solving (119871119875 (119878119896))then 119909119896 isin 119878119896 sube 119878 and 119909119896 is a feasible solution of problem(119876119875) When 120572

119896minus120573119896le 120576 the algorithm terminate From Steps

1 and 6 we have 1198910(119909119896) minus 120573119896le 120576 from the algorithm 120573

119896le V

where V is the global optimal value of problem (119876119875) Because119909119896 is a feasible solution of problem (119876119875) so 1198910(119909119896) ge VThus

V le 1198910 (119909119896) le V + 120576 (13)

Therefore 119909119896 is a 120576-global optimal solution of problem(119876119875)

(b) If the algorithm is infinite then it produces a solutionsequence 119909119896 of problem (119876119875) where for each 119896 ge 1 119909119896 isobtained by solving problem (119871119875 (119878119896)) For each 119878119896 sube 119878 forthe optimal solution119909119896 isin 119878119896 sube 119878 the sequence 119909119896 constitutea solution sequence of problem (119876119875) from the iteration of thealgorithm we have

120573119896le V le 120572

119896= 1198910 (119909119896) 119896 = 1 2 (14)

Because the series 120573119896 do not decrease and have an upper

bound and 120572119896 do not increase and have a lower bound then

the series 120573119896 and 120572

119896 are both convergent Taking the limits

on both sides of (14) we have

lim119896rarrinfin

120573119896le V le lim

119896rarrinfin

120572119896= lim119896rarrinfin

1198910 (119909119896) (15)

Let lim119896rarrinfin

120573119896= 120573 lim

119896rarrinfin120572119896= 120572 then the formula (15)

converts into

120573 le V le lim119896rarrinfin

1198910 (119909119896) = 120572 (16)

Without loss of generality assume that the sequence ofrectangle 119878119896 = [119897119896 119906119896] satisfy 119909119896 isin 119878119896 and 119878119896+1 sub 119878119896

In our algorithm the rectangles are divided into two equalparts continuously then⋂infin

119896=1119878119896+1 = 119909119896 and because of the

continuity of function 1198910(119909)

120573 = V = 120572 = lim119896rarrinfin

1198910 (119909119896) = 1198910 (119909lowast) (17)

So any accumulation of 119909119896 is a global optimal solution ofproblem (119876119875)

5 Numerical Experiment

Several experiments are given to turn out the feasibility andeffectiveness of our algorithm

Example 1

min 11990921+ 11990922

st 0311990911199092ge 1

2 le 1199091le 5

1 le 1199092le 3

(18)

From the algorithm the initial rectangle is 1198781 =

[ 20000 5000010000 30000

] first we solve the problem 119871119875(1198781) its optimalsolution is 1199091 = (20000 30000) and optimal value is1205731= 120573(1198781) = 49996 then 49996 is a lower bound of

the global optimal value of problem (119876119875) Because 1199091 =(20000 30000) is feasible then 119876 = [20000 30000] is a setof current feasible solutions and the current upper boundis 1205721= 1198910(1199091) = 130000 the current optimal solution is

119909lowast = 1199091 = (20000 30000)After that based on our selection rule select the rectangle

with the minimum lower bound 1198781 to divide then 1198781 isdivided into two subrectangles 11987811 = [ 20000 35000

10000 30000] and

11987812 = [ 35000 5000010000 30000

] from the dividing method in Section 31then reduce the rectangles using the reduction technique inSection 32 and the new rectangle after reduction is denotedby 1198782 = 11987811 = [ 20000 35000

10000 30000] Solving the linear relaxation

programming problem 119871119875 on the rectangle 1198782 its optimalvalue is 120573

2= 120573(11987811) = 49996 then the lower bound

of the original problem is not updated also being 49996Next we choose 1198782 to divide until sdot sdot sdot the 15th iteration11987814 = [ 20000 20408

16538 17019] solve the linear relaxation programming

problem 119871119875(11987814) its optimal solution is (20000 16665) andoptimal value is 67765 while the current upper boundis 68151 the current optimal solution is (20000 16778)Because |68151 minus 67765| lt 01 it satisfies our terminationrule then the optimal value of the original problem is 68151the lower bound of the optimal value is 67765 and theoptimal solution is 119909 = (20000 16778) here the lowerbound of the optimal value is also approximate optimal valuewhere the accuracy is 120576 = 01

Table 2 shows the different results of Example 1 underdifferent accuracy

Mathematical Problems in Engineering 5

Table 1

Example The optimal solution within accuracy or one solution among solutions1199091

1199092

1199093

1199094

1199095

1 20000 166672 25576 312793 20000 166674 10000 550005 15000 122476 10156 155947 04267 058798 780000 330001 299958 449998 367753Example Approximate optimal value Iterations CPU (s)1 67778 33 83011282 1183837 49 326965653 67778 29 66764444 10000 9 31621065 minus11629 17 44298066 minus318878 130 540241407 minus33304 20 54109438 10128 98 193921992

Table 2 Different results of Example 1 under different accuracy

Example 1120576 Approximate optimal value Optimal value10119890 minus 2 6777772334392922 6784953802104409

10119890 minus 3 6777777695638590 6778685210977349

10119890 minus 4 6777777810403491 6777777840618791

Example 2

min 611990921+ 411990922+ 511990911199092

st minus611990911199092le minus48

0 le 1199091 1199092le 10

(19)

The optimal value is 1183838

Example 3

min 11990921+ 11990922

st minus0311990911199092le minus1

minus1199091minus 1199092le 1

119909 isin 1198830 = 2 le 1199091le 5 1 le 119909

2le 3

(20)

The optimal value is 67778

Example 4

min 1199091

st 1

41199091+1

21199092minus1

611990922minus1

611990921le 1

1

1411990921+1

1411990922minus3

71199091minus3

71199092le 1

1 le 1199091le 55 1 le 119909

2le 55

(21)

The optimal value is 10000

Example 5

min minus1199091+ 1199091119909052minus 1199092

st minus61199091+ 81199092le 3

31199091minus 1199092le 3

119909 isin 1198830 = 119909 | 0 le 119909119894le 15 119894 = 1 2

(22)

The optimal value is minus11629

Example 6

min 611990921+ 411990922+ 25(119909

1+ 1199092)2minus 25 (10119909

1+ 10119909

2)

st 3(1199091minus 1199092)2minus 3 (10119909

1+ 10119909

2) le minus48

0 le 1199091 1199092le 10

(23)

The optimal value is minus318878

Example 7

min 2111990921+ 34119909

11199092minus 241199092

2+ 21199091minus 14119909

2

st 211990921+ 411990911199092+ 211990922+ 81199091+ 61199092minus 9 le 0

minus511990921minus 811990911199092minus 511990922minus 41199091+ 41199092+ 4 le 0

1199091+ 21199092le 2 119909 isin [0 1]2

(24)

The optimal value is minus33205

6 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

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

Mathematical Problems in Engineering 5

Table 1

Example The optimal solution within accuracy or one solution among solutions1199091

1199092

1199093

1199094

1199095

1 20000 166672 25576 312793 20000 166674 10000 550005 15000 122476 10156 155947 04267 058798 780000 330001 299958 449998 367753Example Approximate optimal value Iterations CPU (s)1 67778 33 83011282 1183837 49 326965653 67778 29 66764444 10000 9 31621065 minus11629 17 44298066 minus318878 130 540241407 minus33304 20 54109438 10128 98 193921992

Table 2 Different results of Example 1 under different accuracy

Example 1120576 Approximate optimal value Optimal value10119890 minus 2 6777772334392922 6784953802104409

10119890 minus 3 6777777695638590 6778685210977349

10119890 minus 4 6777777810403491 6777777840618791

Example 2

min 611990921+ 411990922+ 511990911199092

st minus611990911199092le minus48

0 le 1199091 1199092le 10

(19)

The optimal value is 1183838

Example 3

min 11990921+ 11990922

st minus0311990911199092le minus1

minus1199091minus 1199092le 1

119909 isin 1198830 = 2 le 1199091le 5 1 le 119909

2le 3

(20)

The optimal value is 67778

Example 4

min 1199091

st 1

41199091+1

21199092minus1

611990922minus1

611990921le 1

1

1411990921+1

1411990922minus3

71199091minus3

71199092le 1

1 le 1199091le 55 1 le 119909

2le 55

(21)

The optimal value is 10000

Example 5

min minus1199091+ 1199091119909052minus 1199092

st minus61199091+ 81199092le 3

31199091minus 1199092le 3

119909 isin 1198830 = 119909 | 0 le 119909119894le 15 119894 = 1 2

(22)

The optimal value is minus11629

Example 6

min 611990921+ 411990922+ 25(119909

1+ 1199092)2minus 25 (10119909

1+ 10119909

2)

st 3(1199091minus 1199092)2minus 3 (10119909

1+ 10119909

2) le minus48

0 le 1199091 1199092le 10

(23)

The optimal value is minus318878

Example 7

min 2111990921+ 34119909

11199092minus 241199092

2+ 21199091minus 14119909

2

st 211990921+ 411990911199092+ 211990922+ 81199091+ 61199092minus 9 le 0

minus511990921minus 811990911199092minus 511990922minus 41199091+ 41199092+ 4 le 0

1199091+ 21199092le 2 119909 isin [0 1]2

(24)

The optimal value is minus33205

6 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

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 Mathematical Problems in Engineering

Example 8

min 5357811990923+ 08357119909

11199095+ 372392119909

1

st 2584 times 10minus511990931199095minus 6663 times 10minus3119909

21199095

minus734 times 10minus511990911199094le 1

853007 times 10minus411990921199095+ 9395 times 10minus5119909

11199094

minus3385 times 10minus411990931199095le 1

minus11990921199095minus 042119909

11199092minus 0305861199092

3

le minus13303294 times 103

minus11990931199095minus 02668119909

11199093minus 040584119909

31199094

le minus22751327 times 103

24186 times 10minus411990921199095+ 10159 times 10minus4119909

11199092

+7379 times 10minus511990923le 1

29955 times 10minus411990931199095+ 7992 times 10minus5119909

11199093

+12157 times 10minus411990931199094le 1

119909 isin 1198830 = 119909 | 78 le 1199091le 102 33 le 119909

2le 45 27

le 119909119894le 45 119894 = 3 4 5

(25)

The optimal value is 10128 times 104

We choose 120576 = 10120576 minus 4 then the approximate optimalvalue satisfying accuracy and the CPU running time areobtained the results are shown in Table 1

6 Conclusion

In this paper we presented a branch and bound reducedalgorithm for solving the quadratic programming problemswith quadratic constraints By constructing a linear relaxationprogramming problem the lower bound of the optimalvalue of original problem can be obtained Meanwhile weused a rectangle reduction technique to improve the degreeof approximation and the convergence rate of accelera-tion Numerical experiments show the effectiveness of ouralgorithm

Acknowledgment

Thework is supported by the Foundation of National NaturalScience of China under Grant no 11161001

References

[1] T Van Voorhis ldquoA global optimization algorithm usingLagrangian underestimates and the interval Newton methodrdquoJournal of Global Optimization vol 24 no 3 pp 349ndash370 2002

[2] X J Zheng X L Sun and D Li ldquoConvex relaxations fornonconvex quadratically constrained quadratic programmingmatrix cone decomposition and polyhedral approximationrdquoMathematical Programming B vol 129 no 2 pp 301ndash329 2011

[3] Y Gao H Xue and P Shen ldquoA new rectangle branch-and-reduce approach for solving nonconvex quadratic program-ming problemsrdquo Applied Mathematics and Computation vol168 no 2 pp 1409ndash1418 2005

[4] C Audet P Hansen B Jaumard and G Savard ldquoA branchand cut algorithm for nonconvex quadratically constrained

quadratic programmingrdquoMathematical ProgrammingA vol 87no 1 pp 131ndash152 2000

[5] S-J Qu Y Ji and K-C Zhang ldquoA deterministic globaloptimization algorithm based on a linearizing method fornonconvex quadratically constrained programsrdquoMathematicaland Computer Modelling vol 48 no 11-12 pp 1737ndash1743 2008

[6] H Wu and K Zhang ldquoA new accelerating method for globalnon-convex quadratic optimization with non-convex quadraticconstraintsrdquoAppliedMathematics andComputation vol 197 no2 pp 810ndash818 2008

[7] J Linderoth ldquoA simplicial branch-and-bound algorithm forsolving quadratically constrained quadratic programsrdquo Math-ematical Programming B vol 103 no 2 pp 251ndash282 2005

[8] H Tuy and N T Hoai-Phuong ldquoA robust algorithm forquadratic optimization under quadratic constraintsrdquo Journal ofGlobal Optimization vol 37 no 4 pp 557ndash569 2007

[9] X L Sun J L Li and H Z Luo ldquoConvex relaxation andLagrangian decomposition for indefinite integer quadratic pro-grammingrdquo Optimization vol 59 no 5-6 pp 627ndash641 2010

[10] M Salahi ldquoConvex optimization approach to a single quadrat-ically constrained quadratic minimization problemrdquo CentralEuropean Journal of Operations Research (CEJOR) vol 18 no2 pp 181ndash187 2010

[11] X J Zheng X L Sun and D Li ldquoNonconvex quadraticallyconstrained quadratic programming best DC decompositionsand their SDP representationsrdquo Journal of Global Optimizationvol 50 no 4 pp 695ndash712 2011

[12] X Bao N V Sahinidis and M Tawarmalani ldquoSemidefiniterelaxations for quadratically constrained quadratic program-ming a review and comparisonsrdquo Mathematical ProgrammingB vol 129 no 1 pp 129ndash157 2011

[13] D S Kim N N Tam and N D Yen ldquoSolution existenceand stability of quadratically constrained convex quadraticprogramsrdquoOptimization Letters vol 6 no 2 pp 363ndash373 2012

[14] S Burer and H Dong ldquoRepresenting quadratically constrainedquadratic programs as generalized copositive programsrdquo Oper-ations Research Letters vol 40 no 3 pp 203ndash206 2012

[15] X J Zheng X L Sun D Li and Y F Xu ldquoOn zero dualitygap in nonconvex quadratic programming problemsrdquo Journalof Global Optimization vol 52 no 2 pp 229ndash242 2012

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