Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 423040 8 pageshttpdxdoiorg1011552013423040

Research ArticleA New Gap Function for Vector VariationalInequalities with an Application

Hui-qiang Ma1 Nan-jing Huang2 Meng Wu3 and Donal OrsquoRegan4

1 School of Economics Southwest University for Nationalities Chengdu Sichuan 610041 China2Department of Mathematics Sichuan University Chengdu Sichuan 610064 China3 Business School Sichuan University Chengdu Sichuan 610064 China4 School of Mathematics Statistics and Applied Mathematics National University of Ireland Galway Ireland

Correspondence should be addressed to Nan-jing Huang nanjinghuanghotmailcom

Received 29 May 2013 Accepted 4 June 2013

Academic Editor Gue Myung Lee

Copyright copy 2013 Hui-qiang Ma 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 consider a vector variational inequality in a finite-dimensional space A new gap function is proposed and an equivalentoptimization problem for the vector variational inequality is also provided Under some suitable conditions we prove that thegap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for theequivalent optimization problem solves the vector variational inequality As an application we use the new gap function toreformulate a stochastic vector variational inequality as a deterministic optimization problem We solve this optimization problemby employing the sample average approximation method The convergence of optimal solutions of the approximation problems isalso investigated

1 Introduction

The vector variational inequality (VVI for short) which wasfirst proposed by Giannessi [1] has been widely investigatedby many authors (see [2ndash9] and the references therein) VVIcan be used to model a range of vector equilibrium problemsin economics traffic networks and migration equilibriumproblems (see [1])

One approach for solving a VVI is to transform it intoan equivalent optimization problem by using a gap functionA gap function was first introduced to study optimizationproblems and has become a powerful tool in the studyof convex optimization problems Also a gap function wasintroduced in the study of scalar variational inequalities Itcan reformulate a scalar variational inequality as an equiv-alent optimization problem and so some effective solutionmethods and algorithms for optimization problems can beused to find solutions of variational inequalities Recentlymany authors extended the theory of gap functions to VVIand vector equilibrium problems (see [2 4 6ndash9]) In thispaper we present a new gap function forVVI and reformulate

it as an equivalent optimization problem We also prove thatthe gap function is directionally differentiable and that anypoint satisfying the first-order necessary optimality conditionfor the equivalent optimization problem solves the VVI

In many practical problems problem data will involvesome uncertain factors In order to reflect the uncertain-ties stochastic vector variational inequalities are neededRecently stochastic scalar variational inequalities havereceived a lot of attention in the literature (see [10ndash20])The ERM (expected residual minimization) method wasproposed by Chen and Fukushima [11] in the study ofstochastic complementarity problems They formulated astochastic linear complementarity problem (SLCP) as a min-imization problem which minimizes the expectation of aNCP function (also called a residual function) of SLCP andregarded a solution of theminimization problem as a solutionof SLCP This method is the so-called expected residualminimization method Following the ideas of Chen andFukushima [11] Zhang and Chen [20] considered stochasticnonlinear complementary problems Luo and Lin [18 19]generalized the expected residual minimization method to

2 Journal of Applied Mathematics

solve a stochastic linear andor nonlinear variational inequal-ity problem However in comparison to stochastic scalarvariational inequalities there are very few results in theliterature on stochastic vector variational inequalities In thispaper we consider a deterministic reformulation for thestochastic vector variational inequality (SVVI) Our focus ison the expected residualminimization (ERM)method for thestochastic vector variational inequality It is well known thatVVI is more complicated than a variational inequality andthey model many practical problems Therefore it is mean-ingful and interesting to study stochastic vector variationalinequalities

The rest of this paper is organized as follows In Section 2somepreliminaries are given In Section 3 a newgap functionfor VVI is constructed and some suitable conditions aregiven to ensure that the new gap function is directionallydifferentiable and that any point satisfying the first-ordernecessary condition of optimality for the new optimizationproblem solves the vector variational inequality In Section 4the stochastic VVI is presented and the new gap functionis used to reformulate SVVI as a deterministic optimizationproblem

2 Preliminaries

In this section we will introduce some basic notations andpreliminary results

Throughout this paper denote by 119909119879 the transpose of avector or matrix 119909 by | sdot | the Euclidean norm of a vectoror matrix and by ⟨sdot sdot⟩ the inner product in R119899 Let 119870 bea nonempty closed and convex set of R119899 119865

119894 R119899 rarr R119899

(119894 = 1 119901) mappings and 119865 = (1198651 119865

119901)119879 The vector

variational inequality is to find a vector 119909lowast isin 119870 such that

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(1)

where R119901

+ is the nonnegative orthant of R119901 and int R119901+denotes the interior ofR119901+ Denote by 119878 the solution set of VVI(1) and by 119878

120585the solution set of the following scalar variational

inequality (VI120585) find a vector 119909lowast isin 119870 such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (2)

where 120585 isin 119861 = 120585 isin R119901

+ sum119901

119895=1120585119895= 1

Definition 1 A function 120593120585 119870 rarr R is said to be a gap

function for VI120585(2) if it satisfies the following properties

(i) 120593120585(119909) ge 0 for all 119909 isin 119870

(ii) 120593120585(119909lowast) = 0 iff 119909lowast solves VI

120585(2)

Definition 2 A function 120595 119870 rarr R is said to be a gapfunction for VVI (1) if it satisfies the following properties

(i) 120595(119909) ge 0 for all 119909 isin 119870(ii) 120595(119909lowast) = 0 iff 119909lowast solves VVI (1)

Suppose that 119866 is an 119899 times 119899 symmetric positive definitematrix Let

120593120585 (119909) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(3)

where |119909|2119866= ⟨119909 119866119909⟩ Note that

radic120582min |119909| le |119909|119866 le radic120582max |119909| (4)

where 120582min and 120582max are the smallest and largest eigenvaluesof 119866 respectively It was shown in [21] that the maximum in(3) is attained at

119867120585 (119909) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909)) (5)

where Proj119870119866(119909) is the projection of the point 119909 onto the

closed convex set 119870 with respect to the norm | sdot |119866 Thus

120593120585 (119909) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909) 119909 minus 119867120585 (

119909)⟩ minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909)

10038161003816100381610038161003816

2

119866 (6)

Lemma 3 The projection operator Proj119870119866(sdot) is nonexpansive

that is10038161003816100381610038161003816Proj119870119866(119909) minus Proj

119870119866(119910)

10038161003816100381610038161003816119866le1003816100381610038161003816119909 minus 119910

1003816100381610038161003816119866 forall119909 119910 isin R

119899 (7)

Lemma 4 The function 120593120585is a gap function for VI

120585(2) and

119909lowastisin 119870 solves VI

120585(2) iff it solves the following optimization

problem

min119909isin119870

120593120585 (119909) (8)

The gap function 120593120585is also called the regularized gap

function for VI120585 When 119865

119895(119895 = 1 119901) are continuously

differentiable we have the following results

Lemma 5 If 119865119895(119895 = 1 119901) are continuously differentiable

then 120593120585is also continuously differentiable in 119909 and its gradient

is given by

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(9)

Lemma 6 Assume that 119865119895are continuously differentiable and

that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for all

119909 isin 119870 (119895 = 1 119901) If 119909 is a stationary point of problem (8)that is

⟨nabla119909120593120585 (119909) 119910 minus 119909⟩ ge 0 forall119910 isin 119870 (10)

then it solves VI120585(2)

Journal of Applied Mathematics 3

3 A New Gap Function for VVI andIts Properties

In this section based on the regularized gap function 120593120585for

VI120585(2) we construct a new gap function for VVI (1) and

establish some properties under some mild conditionsLet

120595 (119909) = min120585isin119861

120593120585 (119909) (11)

Before showing that 120595 is a gap function for VVI we firstpresent a useful result

Lemma 7 The following assertion is true

119878 = cup120585isin119861119878120585 (12)

Proof Suppose that 119909lowast isin cup120585isin119861119878120585 Then there exists a 120585 isin 119861

such that 119909lowast isin 119878120585and

119901

sum

119895=1

120585119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ = ⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0

forall119910 isin 119870

(13)

For any fixed 119910 isin 119870 since 120585 isin 119861 there exists a 119895 isin 1 119901such that

⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (14)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

(15)

Thus we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(16)

This implies that 119909lowast isin 119878 and cup120585isin119861119878120585sub 119878

Conversely suppose that 119909lowast isin 119878 Then we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(17)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

forall119910 isin 119870

cap (minus int R119901+) = 0

(18)

Since 119870 is convex from Theorems 111 and 113 of [22] itfollows that

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ (19)

where 119887 = (1198871 119887

119901) = 0 Moreover we have 119887 gt 0 In fact if

119887119895lt 0 for some 119895 isin 1 119901 then we have

sup119910isin(minus int R119901+)

⟨119887 119910⟩ = +infin (20)

On the other hand

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

le

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119909lowastminus 119909lowast⟩

= 0

lt sup119910isin(minus int R119901+)

⟨119887 119910⟩

(21)

which is a contradiction Thus 119887 gt 0 This implies that forany 119911 isin 119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119911 minus 119909

lowast⟩

ge inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ = 0

(22)

Taking 120585 = 119887(sum119901119895=1119887119895) then 120585 isin 119861 and

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119911 minus 119909

lowast⟩ ge 0 forall119911 isin 119870 (23)

which implies that 119909lowast isin 119878120585and 119878 sub cup

120585isin119861119878120585

This completes the proof

Now we can prove that 120595 is a gap function for VVI (1)

Theorem 8 The function 120595 given by (11) is a gap function forVVI (1) Hence 119909lowast isin 119870 solves VVI (1) iff it solves the followingoptimization problem

min119909isin119870

120595 (119909) (24)

Proof Note that for any 120585 isin 119861 120593120585(119909) given by (3) is a gap

function for VI120585(2) It follows from Definition 1 that 120593

120585(119909) ge

0 for all 119909 isin 119870 and hence 120595(119909) = min120585isin119861

120593120585(119909) ge 0 for all

119909 isin 119870Assume that 120595(119909lowast) = 0 for some 119909lowast isin 119870 From (6) it is

easy to see that 120593120585(119909lowast) is continuous in 120585 Since 119861 is a closed

4 Journal of Applied Mathematics

and bounded set there exists a vector 120585 isin 119861 such that120595(119909lowast) =120593120585(119909lowast) which implies that 120593

120585(119909lowast) = 0 and 119909lowast isin 119878

120585 It follows

from Lemma 7 that 119909lowast solves VVI (1)Suppose that 119909lowast solves VVI (1) From Lemma 7 it follows

that there exists a vector 120585 isin 119861 such that 119909lowast isin 119878120585and so

120593120585(119909lowast) = 0 Since for all 120585 isin 119861 120593

120585(119909lowast) ge 0 we have 120595(119909lowast) =

min120585isin119861

120593120585(119909lowast) = 0

Thus (11) is a gap function for VVI (1) The last assertionis obvious from the definition of gap function

This completes the proof

Since 120595 is constructed based on the regularized gapfunction for VI

120585 we wish to call it a regularized gap function

forVVI (1)Theorem 8 indicates that in order to get a solutionof VVI (1) we only need to solve problem (24) In whatfollows wewill discuss some properties of the regularized gapfunction 120595

Theorem 9 If 119865119895(119895 = 1 119901) are continuously differen-

tiable then the regularized gap function 120595 is directionallydifferentiable in any direction 119889 isin R119899 and its directionalderivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (25)

where 119861(119909) = 120585 isin 119861 120593120585(119909) = min

120585isin119861120593120585(119909)

Proof It follows since the projection operator is nonexpan-sive that 120593

120585(119909) is continuous in (120585 119909)Thus 119861(119909) is nonempty

for any 119909 isin 119870 From Lemma 5 it follows that 120593120585(119909) is

continuously differentiable in 119909 and that

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(26)

is continuous in (120585 119909) It follows fromTheorem 1 of [23] that120595 is directionally differentiable in any direction 119889 isin R119899 andits directional derivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (27)

This completes the proof

By the directional differentiability of 120595 shown inTheorem 9 the first-order necessary condition of optimalityfor problem (24) can be stated as

1205951015840(119909 119910 minus 119909) ge 0 forall119910 isin 119870 (28)

If one wishes to solve VVI (1) via the optimization problem(24) we need to obtain its global optimal solution Howeversince the function 120595 is in general nonconvex it is difficult tofind a global optimal solution Fortunately we can prove thatany point satisfying the first-order condition of optimalitybecomes a solution of VVI (1) Thus existing optimizationalgorithms can be used to find a solution of VVI (1)

Theorem 10 Assume that 119865119895are continuously differentiable

and that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for

all 119909 isin 119870 (119895 = 1 119901) If 119909lowast isin 119870 satisfies the first-ordercondition of optimality (28) then 119909lowast solves VVI (1)

Proof It follows fromTheorem 9 and (28) that for some 120585 isin119861(119909lowast)

⟨nablalowast

119909120593120585(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (29)

holds for any 119910 isin 119870 This implies that 119909lowast is a stationary pointof problem (8) It follows from Lemma 6 that 119909lowast solves VI

120585

FromLemma 7 we see that119909lowast solves VVIThis completes theproof

From Theorems 8 and 10 it is easy to get the followingcorollary

Corollary 11 Assume that the conditions in Theorem 10 areall satisfied If 119909lowast isin 119870 satisfies the first-order condition ofoptimality (28) then 119909lowast is a global optimal solution of problem(24)

4 Stochastic Vector Variational Inequality

In this section we consider the stochastic vector variationalinequality (SVVI) First we present a deterministic refor-mulation for SVVI by employing the ERM method and theregularized gap function Second we solve this reformulationby the SAA method

In most important practical applications the functions119865119895(119895 = 1 119901) always involve some random factors or

uncertainties Let (ΩF 119875) be a probability space Takingthe randomness into account we get a stochastic vectorvariational inequality problem (SVVI) find a vector 119909lowast isin 119870such that

(⟨1198651(119909lowast 120596) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast 120596) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+ forall119910 isin 119870 as

(30)

where 119865119895 R119899 times Ω rarr R119899 are mappings and as is the

abbreviation for ldquoalmost surelyrdquo under the given probabilitymeasure 119875

Because of the random element 120596 we cannot generallyfind a vector 119909lowast isin 119870 such that (30) holds almost surelyThat is (30) is not well defined if we think of solving(30) before knowing the realization 120596 Therefore in orderto get a reasonable resolution an appropriate deterministicreformulation for SVVI becomes an important issue in thestudy of the considered problem In this section we willemploy the ERMmethod to solve (30)

Define120595 (119909 120596) = min

120585isin119861

120593120585 (119909 120596) (31)

where

120593120585 (119909 120596) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(32)

Journal of Applied Mathematics 5

The maximum in (32) is attained at

119867120585 (119909 120596) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909 120596)) (33)

120593120585 (119909 120596) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

(34)

The ERM reformulation is given as follows

min119909isin119870

Θ (119909) = E [120595 (119909 120596)] (35)

where E is the expectation operatorNote that the objective functionΘ(119909) contains the math-

ematical expectation In practical applications it is in generalvery difficult to calculate E[120595(119909 120596)] in a closed form Thuswe will have to approximate it through discretization Oneof the most popular discretization approaches is the sampleaverage approximation method In general for an integrablefunction 120601 Ω rarr R we approximate the expected valueE[120601(120596)] with the sample average (1119873

119896) sum120596119894isinΩ119896

120601(120596119894) where

1205961 120596

119873119896are independently and identically distributed

random samples of 120596 and Ω119896= 1205961 120596

119873119896 By the strong

law of large numbers we get the following lemma

Lemma 12 If 120601(120596) is integrable then

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120601 (120596119894) = E [120601 (120596)] (36)

holds with probability one

Let Θ119896(119909) = (1119873

119896) sum120596119894isinΩ119896

120595(119909 120596119894) Applying the above

technique we get the following approximation of (35)

min119909isin119870

Θ119896 (119909) (37)

In the rest of this section we focus on the case

119865119895 (119909 120596) = 119872119895 (

120596) 119909 + 119876119895 (120596) (38)

(119895 = 1 119901) where119872119895 Ω rarr R119899times119899 and 119876

119895 Ω rarr R119899 are

measurable functions such that

E [10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816

2

] lt +infin E [10038161003816100381610038161003816119876(120596)119895

10038161003816100381610038161003816

2

] lt +infin

119895 = 1 119901

(39)

This condition implies that

E[

[

119901

sum

119895=1

119872119895 (120596)]

]

2

lt +infin

E[

[

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin

(40)

and for any scalar 119888

E[

[

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816119888 +

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin (41)

The following results will be useful in the proof of theconvergence result

Lemma 13 Let 119891 119892 R119901 rarr R+

be continuous Ifmin120585isin119861

119891(120585) lt +infin andmin120585isin119861

119892(120585) lt +infin then

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816 (42)

Proof Without loss of generality we assume thatmin120585isin119861119891(120585) le min

120585isin119861119892(120585) Let 120585minimize 119891 and 120585minimize

119892 respectively Hence 119891(120585) le 119892(120585) and 119892(120585) le 119892(120585) Thus

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

= 119892 (120585) minus 119891 (120585)

le 119892 (120585) minus 119891 (120585)

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816

(43)

This completes the proof

Lemma 14 When 119865119895(119909 120596) = 119872

119895(120596)119909 +119876

119895(120596) (119895 = 1 119901)

the function 120593120585(119909 120596) is continuously differentiable in 119909 almost

surely and its gradient is given by

nabla119909120593120585 (119909 120596) =

119901

sum

119895=1

120585119895119865119895 (119909 120596)

minus (

119901

sum

119895=1

120585119895119872119895 (120596) minus 119866)(119867120585 (

119909 120596) minus 119909)

(44)

Proof Theproof is the same as that ofTheorem 32 in [21] sowe omit it here

Lemma 15 For any 119909 isin 119870 one has

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (45)

Proof For any fixed120596 isin Ω 120593120585(119909 120596) is the gap function of the

following scalar variational inequality find a vector 119909lowast isin 119870such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast 120596) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (46)

6 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

solve a stochastic linear andor nonlinear variational inequal-ity problem However in comparison to stochastic scalarvariational inequalities there are very few results in theliterature on stochastic vector variational inequalities In thispaper we consider a deterministic reformulation for thestochastic vector variational inequality (SVVI) Our focus ison the expected residualminimization (ERM)method for thestochastic vector variational inequality It is well known thatVVI is more complicated than a variational inequality andthey model many practical problems Therefore it is mean-ingful and interesting to study stochastic vector variationalinequalities

The rest of this paper is organized as follows In Section 2somepreliminaries are given In Section 3 a newgap functionfor VVI is constructed and some suitable conditions aregiven to ensure that the new gap function is directionallydifferentiable and that any point satisfying the first-ordernecessary condition of optimality for the new optimizationproblem solves the vector variational inequality In Section 4the stochastic VVI is presented and the new gap functionis used to reformulate SVVI as a deterministic optimizationproblem

2 Preliminaries

In this section we will introduce some basic notations andpreliminary results

Throughout this paper denote by 119909119879 the transpose of avector or matrix 119909 by | sdot | the Euclidean norm of a vectoror matrix and by ⟨sdot sdot⟩ the inner product in R119899 Let 119870 bea nonempty closed and convex set of R119899 119865

119894 R119899 rarr R119899

(119894 = 1 119901) mappings and 119865 = (1198651 119865

119901)119879 The vector

variational inequality is to find a vector 119909lowast isin 119870 such that

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(1)

where R119901

+ is the nonnegative orthant of R119901 and int R119901+denotes the interior ofR119901+ Denote by 119878 the solution set of VVI(1) and by 119878

120585the solution set of the following scalar variational

inequality (VI120585) find a vector 119909lowast isin 119870 such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (2)

where 120585 isin 119861 = 120585 isin R119901

+ sum119901

119895=1120585119895= 1

Definition 1 A function 120593120585 119870 rarr R is said to be a gap

function for VI120585(2) if it satisfies the following properties

(i) 120593120585(119909) ge 0 for all 119909 isin 119870

(ii) 120593120585(119909lowast) = 0 iff 119909lowast solves VI

120585(2)

Definition 2 A function 120595 119870 rarr R is said to be a gapfunction for VVI (1) if it satisfies the following properties

(i) 120595(119909) ge 0 for all 119909 isin 119870(ii) 120595(119909lowast) = 0 iff 119909lowast solves VVI (1)

Suppose that 119866 is an 119899 times 119899 symmetric positive definitematrix Let

120593120585 (119909) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(3)

where |119909|2119866= ⟨119909 119866119909⟩ Note that

radic120582min |119909| le |119909|119866 le radic120582max |119909| (4)

where 120582min and 120582max are the smallest and largest eigenvaluesof 119866 respectively It was shown in [21] that the maximum in(3) is attained at

119867120585 (119909) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909)) (5)

where Proj119870119866(119909) is the projection of the point 119909 onto the

closed convex set 119870 with respect to the norm | sdot |119866 Thus

120593120585 (119909) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909) 119909 minus 119867120585 (

119909)⟩ minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909)

10038161003816100381610038161003816

2

119866 (6)

Lemma 3 The projection operator Proj119870119866(sdot) is nonexpansive

that is10038161003816100381610038161003816Proj119870119866(119909) minus Proj

119870119866(119910)

10038161003816100381610038161003816119866le1003816100381610038161003816119909 minus 119910

1003816100381610038161003816119866 forall119909 119910 isin R

119899 (7)

Lemma 4 The function 120593120585is a gap function for VI

120585(2) and

119909lowastisin 119870 solves VI

120585(2) iff it solves the following optimization

problem

min119909isin119870

120593120585 (119909) (8)

The gap function 120593120585is also called the regularized gap

function for VI120585 When 119865

119895(119895 = 1 119901) are continuously

differentiable we have the following results

Lemma 5 If 119865119895(119895 = 1 119901) are continuously differentiable

then 120593120585is also continuously differentiable in 119909 and its gradient

is given by

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(9)

Lemma 6 Assume that 119865119895are continuously differentiable and

that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for all

119909 isin 119870 (119895 = 1 119901) If 119909 is a stationary point of problem (8)that is

⟨nabla119909120593120585 (119909) 119910 minus 119909⟩ ge 0 forall119910 isin 119870 (10)

then it solves VI120585(2)

Journal of Applied Mathematics 3

3 A New Gap Function for VVI andIts Properties

In this section based on the regularized gap function 120593120585for

VI120585(2) we construct a new gap function for VVI (1) and

establish some properties under some mild conditionsLet

120595 (119909) = min120585isin119861

120593120585 (119909) (11)

Before showing that 120595 is a gap function for VVI we firstpresent a useful result

Lemma 7 The following assertion is true

119878 = cup120585isin119861119878120585 (12)

Proof Suppose that 119909lowast isin cup120585isin119861119878120585 Then there exists a 120585 isin 119861

such that 119909lowast isin 119878120585and

119901

sum

119895=1

120585119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ = ⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0

forall119910 isin 119870

(13)

For any fixed 119910 isin 119870 since 120585 isin 119861 there exists a 119895 isin 1 119901such that

⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (14)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

(15)

Thus we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(16)

This implies that 119909lowast isin 119878 and cup120585isin119861119878120585sub 119878

Conversely suppose that 119909lowast isin 119878 Then we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(17)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

forall119910 isin 119870

cap (minus int R119901+) = 0

(18)

Since 119870 is convex from Theorems 111 and 113 of [22] itfollows that

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ (19)

where 119887 = (1198871 119887

119901) = 0 Moreover we have 119887 gt 0 In fact if

119887119895lt 0 for some 119895 isin 1 119901 then we have

sup119910isin(minus int R119901+)

⟨119887 119910⟩ = +infin (20)

On the other hand

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

le

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119909lowastminus 119909lowast⟩

= 0

lt sup119910isin(minus int R119901+)

⟨119887 119910⟩

(21)

which is a contradiction Thus 119887 gt 0 This implies that forany 119911 isin 119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119911 minus 119909

lowast⟩

ge inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ = 0

(22)

Taking 120585 = 119887(sum119901119895=1119887119895) then 120585 isin 119861 and

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119911 minus 119909

lowast⟩ ge 0 forall119911 isin 119870 (23)

which implies that 119909lowast isin 119878120585and 119878 sub cup

120585isin119861119878120585

This completes the proof

Now we can prove that 120595 is a gap function for VVI (1)

Theorem 8 The function 120595 given by (11) is a gap function forVVI (1) Hence 119909lowast isin 119870 solves VVI (1) iff it solves the followingoptimization problem

min119909isin119870

120595 (119909) (24)

Proof Note that for any 120585 isin 119861 120593120585(119909) given by (3) is a gap

function for VI120585(2) It follows from Definition 1 that 120593

120585(119909) ge

0 for all 119909 isin 119870 and hence 120595(119909) = min120585isin119861

120593120585(119909) ge 0 for all

119909 isin 119870Assume that 120595(119909lowast) = 0 for some 119909lowast isin 119870 From (6) it is

easy to see that 120593120585(119909lowast) is continuous in 120585 Since 119861 is a closed

4 Journal of Applied Mathematics

and bounded set there exists a vector 120585 isin 119861 such that120595(119909lowast) =120593120585(119909lowast) which implies that 120593

120585(119909lowast) = 0 and 119909lowast isin 119878

120585 It follows

from Lemma 7 that 119909lowast solves VVI (1)Suppose that 119909lowast solves VVI (1) From Lemma 7 it follows

that there exists a vector 120585 isin 119861 such that 119909lowast isin 119878120585and so

120593120585(119909lowast) = 0 Since for all 120585 isin 119861 120593

120585(119909lowast) ge 0 we have 120595(119909lowast) =

min120585isin119861

120593120585(119909lowast) = 0

Thus (11) is a gap function for VVI (1) The last assertionis obvious from the definition of gap function

This completes the proof

Since 120595 is constructed based on the regularized gapfunction for VI

120585 we wish to call it a regularized gap function

forVVI (1)Theorem 8 indicates that in order to get a solutionof VVI (1) we only need to solve problem (24) In whatfollows wewill discuss some properties of the regularized gapfunction 120595

Theorem 9 If 119865119895(119895 = 1 119901) are continuously differen-

tiable then the regularized gap function 120595 is directionallydifferentiable in any direction 119889 isin R119899 and its directionalderivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (25)

where 119861(119909) = 120585 isin 119861 120593120585(119909) = min

120585isin119861120593120585(119909)

Proof It follows since the projection operator is nonexpan-sive that 120593

120585(119909) is continuous in (120585 119909)Thus 119861(119909) is nonempty

for any 119909 isin 119870 From Lemma 5 it follows that 120593120585(119909) is

continuously differentiable in 119909 and that

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(26)

is continuous in (120585 119909) It follows fromTheorem 1 of [23] that120595 is directionally differentiable in any direction 119889 isin R119899 andits directional derivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (27)

This completes the proof

By the directional differentiability of 120595 shown inTheorem 9 the first-order necessary condition of optimalityfor problem (24) can be stated as

1205951015840(119909 119910 minus 119909) ge 0 forall119910 isin 119870 (28)

If one wishes to solve VVI (1) via the optimization problem(24) we need to obtain its global optimal solution Howeversince the function 120595 is in general nonconvex it is difficult tofind a global optimal solution Fortunately we can prove thatany point satisfying the first-order condition of optimalitybecomes a solution of VVI (1) Thus existing optimizationalgorithms can be used to find a solution of VVI (1)

Theorem 10 Assume that 119865119895are continuously differentiable

and that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for

all 119909 isin 119870 (119895 = 1 119901) If 119909lowast isin 119870 satisfies the first-ordercondition of optimality (28) then 119909lowast solves VVI (1)

Proof It follows fromTheorem 9 and (28) that for some 120585 isin119861(119909lowast)

⟨nablalowast

119909120593120585(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (29)

holds for any 119910 isin 119870 This implies that 119909lowast is a stationary pointof problem (8) It follows from Lemma 6 that 119909lowast solves VI

120585

FromLemma 7 we see that119909lowast solves VVIThis completes theproof

From Theorems 8 and 10 it is easy to get the followingcorollary

Corollary 11 Assume that the conditions in Theorem 10 areall satisfied If 119909lowast isin 119870 satisfies the first-order condition ofoptimality (28) then 119909lowast is a global optimal solution of problem(24)

4 Stochastic Vector Variational Inequality

In this section we consider the stochastic vector variationalinequality (SVVI) First we present a deterministic refor-mulation for SVVI by employing the ERM method and theregularized gap function Second we solve this reformulationby the SAA method

In most important practical applications the functions119865119895(119895 = 1 119901) always involve some random factors or

uncertainties Let (ΩF 119875) be a probability space Takingthe randomness into account we get a stochastic vectorvariational inequality problem (SVVI) find a vector 119909lowast isin 119870such that

(⟨1198651(119909lowast 120596) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast 120596) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+ forall119910 isin 119870 as

(30)

where 119865119895 R119899 times Ω rarr R119899 are mappings and as is the

abbreviation for ldquoalmost surelyrdquo under the given probabilitymeasure 119875

Because of the random element 120596 we cannot generallyfind a vector 119909lowast isin 119870 such that (30) holds almost surelyThat is (30) is not well defined if we think of solving(30) before knowing the realization 120596 Therefore in orderto get a reasonable resolution an appropriate deterministicreformulation for SVVI becomes an important issue in thestudy of the considered problem In this section we willemploy the ERMmethod to solve (30)

Define120595 (119909 120596) = min

120585isin119861

120593120585 (119909 120596) (31)

where

120593120585 (119909 120596) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(32)

Journal of Applied Mathematics 5

The maximum in (32) is attained at

119867120585 (119909 120596) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909 120596)) (33)

120593120585 (119909 120596) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

(34)

The ERM reformulation is given as follows

min119909isin119870

Θ (119909) = E [120595 (119909 120596)] (35)

where E is the expectation operatorNote that the objective functionΘ(119909) contains the math-

ematical expectation In practical applications it is in generalvery difficult to calculate E[120595(119909 120596)] in a closed form Thuswe will have to approximate it through discretization Oneof the most popular discretization approaches is the sampleaverage approximation method In general for an integrablefunction 120601 Ω rarr R we approximate the expected valueE[120601(120596)] with the sample average (1119873

119896) sum120596119894isinΩ119896

120601(120596119894) where

1205961 120596

119873119896are independently and identically distributed

random samples of 120596 and Ω119896= 1205961 120596

119873119896 By the strong

law of large numbers we get the following lemma

Lemma 12 If 120601(120596) is integrable then

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120601 (120596119894) = E [120601 (120596)] (36)

holds with probability one

Let Θ119896(119909) = (1119873

119896) sum120596119894isinΩ119896

120595(119909 120596119894) Applying the above

technique we get the following approximation of (35)

min119909isin119870

Θ119896 (119909) (37)

In the rest of this section we focus on the case

119865119895 (119909 120596) = 119872119895 (

120596) 119909 + 119876119895 (120596) (38)

(119895 = 1 119901) where119872119895 Ω rarr R119899times119899 and 119876

119895 Ω rarr R119899 are

measurable functions such that

E [10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816

2

] lt +infin E [10038161003816100381610038161003816119876(120596)119895

10038161003816100381610038161003816

2

] lt +infin

119895 = 1 119901

(39)

This condition implies that

E[

[

119901

sum

119895=1

119872119895 (120596)]

]

2

lt +infin

E[

[

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin

(40)

and for any scalar 119888

E[

[

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816119888 +

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin (41)

The following results will be useful in the proof of theconvergence result

Lemma 13 Let 119891 119892 R119901 rarr R+

be continuous Ifmin120585isin119861

119891(120585) lt +infin andmin120585isin119861

119892(120585) lt +infin then

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816 (42)

Proof Without loss of generality we assume thatmin120585isin119861119891(120585) le min

120585isin119861119892(120585) Let 120585minimize 119891 and 120585minimize

119892 respectively Hence 119891(120585) le 119892(120585) and 119892(120585) le 119892(120585) Thus

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

= 119892 (120585) minus 119891 (120585)

le 119892 (120585) minus 119891 (120585)

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816

(43)

This completes the proof

Lemma 14 When 119865119895(119909 120596) = 119872

119895(120596)119909 +119876

119895(120596) (119895 = 1 119901)

the function 120593120585(119909 120596) is continuously differentiable in 119909 almost

surely and its gradient is given by

nabla119909120593120585 (119909 120596) =

119901

sum

119895=1

120585119895119865119895 (119909 120596)

minus (

119901

sum

119895=1

120585119895119872119895 (120596) minus 119866)(119867120585 (

119909 120596) minus 119909)

(44)

Proof Theproof is the same as that ofTheorem 32 in [21] sowe omit it here

Lemma 15 For any 119909 isin 119870 one has

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (45)

Proof For any fixed120596 isin Ω 120593120585(119909 120596) is the gap function of the

following scalar variational inequality find a vector 119909lowast isin 119870such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast 120596) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (46)

6 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

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

Journal of Applied Mathematics 3

3 A New Gap Function for VVI andIts Properties

In this section based on the regularized gap function 120593120585for

VI120585(2) we construct a new gap function for VVI (1) and

establish some properties under some mild conditionsLet

120595 (119909) = min120585isin119861

120593120585 (119909) (11)

Before showing that 120595 is a gap function for VVI we firstpresent a useful result

Lemma 7 The following assertion is true

119878 = cup120585isin119861119878120585 (12)

Proof Suppose that 119909lowast isin cup120585isin119861119878120585 Then there exists a 120585 isin 119861

such that 119909lowast isin 119878120585and

119901

sum

119895=1

120585119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ = ⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0

forall119910 isin 119870

(13)

For any fixed 119910 isin 119870 since 120585 isin 119861 there exists a 119895 isin 1 119901such that

⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (14)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

(15)

Thus we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(16)

This implies that 119909lowast isin 119878 and cup120585isin119861119878120585sub 119878

Conversely suppose that 119909lowast isin 119878 Then we have

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+

forall119910 isin 119870

(17)

and so

(⟨1198651(119909lowast) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast) 119910 minus 119909

lowast⟩)

119879

forall119910 isin 119870

cap (minus int R119901+) = 0

(18)

Since 119870 is convex from Theorems 111 and 113 of [22] itfollows that

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ (19)

where 119887 = (1198871 119887

119901) = 0 Moreover we have 119887 gt 0 In fact if

119887119895lt 0 for some 119895 isin 1 119901 then we have

sup119910isin(minus int R119901+)

⟨119887 119910⟩ = +infin (20)

On the other hand

inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

le

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119909lowastminus 119909lowast⟩

= 0

lt sup119910isin(minus int R119901+)

⟨119887 119910⟩

(21)

which is a contradiction Thus 119887 gt 0 This implies that forany 119911 isin 119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119911 minus 119909

lowast⟩

ge inf119910isin119870

119901

sum

119895=1

119887119895⟨119865119895(119909lowast) 119910 minus 119909

lowast⟩

ge sup119910isin(minus int R119901+)

⟨119887 119910⟩ = 0

(22)

Taking 120585 = 119887(sum119901119895=1119887119895) then 120585 isin 119861 and

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast) 119911 minus 119909

lowast⟩ ge 0 forall119911 isin 119870 (23)

which implies that 119909lowast isin 119878120585and 119878 sub cup

120585isin119861119878120585

This completes the proof

Now we can prove that 120595 is a gap function for VVI (1)

Theorem 8 The function 120595 given by (11) is a gap function forVVI (1) Hence 119909lowast isin 119870 solves VVI (1) iff it solves the followingoptimization problem

min119909isin119870

120595 (119909) (24)

Proof Note that for any 120585 isin 119861 120593120585(119909) given by (3) is a gap

function for VI120585(2) It follows from Definition 1 that 120593

120585(119909) ge

0 for all 119909 isin 119870 and hence 120595(119909) = min120585isin119861

120593120585(119909) ge 0 for all

119909 isin 119870Assume that 120595(119909lowast) = 0 for some 119909lowast isin 119870 From (6) it is

easy to see that 120593120585(119909lowast) is continuous in 120585 Since 119861 is a closed

4 Journal of Applied Mathematics

and bounded set there exists a vector 120585 isin 119861 such that120595(119909lowast) =120593120585(119909lowast) which implies that 120593

120585(119909lowast) = 0 and 119909lowast isin 119878

120585 It follows

from Lemma 7 that 119909lowast solves VVI (1)Suppose that 119909lowast solves VVI (1) From Lemma 7 it follows

that there exists a vector 120585 isin 119861 such that 119909lowast isin 119878120585and so

120593120585(119909lowast) = 0 Since for all 120585 isin 119861 120593

120585(119909lowast) ge 0 we have 120595(119909lowast) =

min120585isin119861

120593120585(119909lowast) = 0

Thus (11) is a gap function for VVI (1) The last assertionis obvious from the definition of gap function

This completes the proof

Since 120595 is constructed based on the regularized gapfunction for VI

120585 we wish to call it a regularized gap function

forVVI (1)Theorem 8 indicates that in order to get a solutionof VVI (1) we only need to solve problem (24) In whatfollows wewill discuss some properties of the regularized gapfunction 120595

Theorem 9 If 119865119895(119895 = 1 119901) are continuously differen-

tiable then the regularized gap function 120595 is directionallydifferentiable in any direction 119889 isin R119899 and its directionalderivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (25)

where 119861(119909) = 120585 isin 119861 120593120585(119909) = min

120585isin119861120593120585(119909)

Proof It follows since the projection operator is nonexpan-sive that 120593

120585(119909) is continuous in (120585 119909)Thus 119861(119909) is nonempty

for any 119909 isin 119870 From Lemma 5 it follows that 120593120585(119909) is

continuously differentiable in 119909 and that

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(26)

is continuous in (120585 119909) It follows fromTheorem 1 of [23] that120595 is directionally differentiable in any direction 119889 isin R119899 andits directional derivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (27)

This completes the proof

By the directional differentiability of 120595 shown inTheorem 9 the first-order necessary condition of optimalityfor problem (24) can be stated as

1205951015840(119909 119910 minus 119909) ge 0 forall119910 isin 119870 (28)

If one wishes to solve VVI (1) via the optimization problem(24) we need to obtain its global optimal solution Howeversince the function 120595 is in general nonconvex it is difficult tofind a global optimal solution Fortunately we can prove thatany point satisfying the first-order condition of optimalitybecomes a solution of VVI (1) Thus existing optimizationalgorithms can be used to find a solution of VVI (1)

Theorem 10 Assume that 119865119895are continuously differentiable

and that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for

all 119909 isin 119870 (119895 = 1 119901) If 119909lowast isin 119870 satisfies the first-ordercondition of optimality (28) then 119909lowast solves VVI (1)

Proof It follows fromTheorem 9 and (28) that for some 120585 isin119861(119909lowast)

⟨nablalowast

119909120593120585(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (29)

holds for any 119910 isin 119870 This implies that 119909lowast is a stationary pointof problem (8) It follows from Lemma 6 that 119909lowast solves VI

120585

FromLemma 7 we see that119909lowast solves VVIThis completes theproof

From Theorems 8 and 10 it is easy to get the followingcorollary

Corollary 11 Assume that the conditions in Theorem 10 areall satisfied If 119909lowast isin 119870 satisfies the first-order condition ofoptimality (28) then 119909lowast is a global optimal solution of problem(24)

4 Stochastic Vector Variational Inequality

In this section we consider the stochastic vector variationalinequality (SVVI) First we present a deterministic refor-mulation for SVVI by employing the ERM method and theregularized gap function Second we solve this reformulationby the SAA method

In most important practical applications the functions119865119895(119895 = 1 119901) always involve some random factors or

uncertainties Let (ΩF 119875) be a probability space Takingthe randomness into account we get a stochastic vectorvariational inequality problem (SVVI) find a vector 119909lowast isin 119870such that

(⟨1198651(119909lowast 120596) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast 120596) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+ forall119910 isin 119870 as

(30)

where 119865119895 R119899 times Ω rarr R119899 are mappings and as is the

abbreviation for ldquoalmost surelyrdquo under the given probabilitymeasure 119875

Because of the random element 120596 we cannot generallyfind a vector 119909lowast isin 119870 such that (30) holds almost surelyThat is (30) is not well defined if we think of solving(30) before knowing the realization 120596 Therefore in orderto get a reasonable resolution an appropriate deterministicreformulation for SVVI becomes an important issue in thestudy of the considered problem In this section we willemploy the ERMmethod to solve (30)

Define120595 (119909 120596) = min

120585isin119861

120593120585 (119909 120596) (31)

where

120593120585 (119909 120596) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(32)

Journal of Applied Mathematics 5

The maximum in (32) is attained at

119867120585 (119909 120596) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909 120596)) (33)

120593120585 (119909 120596) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

(34)

The ERM reformulation is given as follows

min119909isin119870

Θ (119909) = E [120595 (119909 120596)] (35)

where E is the expectation operatorNote that the objective functionΘ(119909) contains the math-

ematical expectation In practical applications it is in generalvery difficult to calculate E[120595(119909 120596)] in a closed form Thuswe will have to approximate it through discretization Oneof the most popular discretization approaches is the sampleaverage approximation method In general for an integrablefunction 120601 Ω rarr R we approximate the expected valueE[120601(120596)] with the sample average (1119873

119896) sum120596119894isinΩ119896

120601(120596119894) where

1205961 120596

119873119896are independently and identically distributed

random samples of 120596 and Ω119896= 1205961 120596

119873119896 By the strong

law of large numbers we get the following lemma

Lemma 12 If 120601(120596) is integrable then

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120601 (120596119894) = E [120601 (120596)] (36)

holds with probability one

Let Θ119896(119909) = (1119873

119896) sum120596119894isinΩ119896

120595(119909 120596119894) Applying the above

technique we get the following approximation of (35)

min119909isin119870

Θ119896 (119909) (37)

In the rest of this section we focus on the case

119865119895 (119909 120596) = 119872119895 (

120596) 119909 + 119876119895 (120596) (38)

(119895 = 1 119901) where119872119895 Ω rarr R119899times119899 and 119876

119895 Ω rarr R119899 are

measurable functions such that

E [10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816

2

] lt +infin E [10038161003816100381610038161003816119876(120596)119895

10038161003816100381610038161003816

2

] lt +infin

119895 = 1 119901

(39)

This condition implies that

E[

[

119901

sum

119895=1

119872119895 (120596)]

]

2

lt +infin

E[

[

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin

(40)

and for any scalar 119888

E[

[

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816119888 +

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin (41)

The following results will be useful in the proof of theconvergence result

Lemma 13 Let 119891 119892 R119901 rarr R+

be continuous Ifmin120585isin119861

119891(120585) lt +infin andmin120585isin119861

119892(120585) lt +infin then

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816 (42)

Proof Without loss of generality we assume thatmin120585isin119861119891(120585) le min

120585isin119861119892(120585) Let 120585minimize 119891 and 120585minimize

119892 respectively Hence 119891(120585) le 119892(120585) and 119892(120585) le 119892(120585) Thus

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

= 119892 (120585) minus 119891 (120585)

le 119892 (120585) minus 119891 (120585)

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816

(43)

This completes the proof

Lemma 14 When 119865119895(119909 120596) = 119872

119895(120596)119909 +119876

119895(120596) (119895 = 1 119901)

the function 120593120585(119909 120596) is continuously differentiable in 119909 almost

surely and its gradient is given by

nabla119909120593120585 (119909 120596) =

119901

sum

119895=1

120585119895119865119895 (119909 120596)

minus (

119901

sum

119895=1

120585119895119872119895 (120596) minus 119866)(119867120585 (

119909 120596) minus 119909)

(44)

Proof Theproof is the same as that ofTheorem 32 in [21] sowe omit it here

Lemma 15 For any 119909 isin 119870 one has

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (45)

Proof For any fixed120596 isin Ω 120593120585(119909 120596) is the gap function of the

following scalar variational inequality find a vector 119909lowast isin 119870such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast 120596) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (46)

6 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

and bounded set there exists a vector 120585 isin 119861 such that120595(119909lowast) =120593120585(119909lowast) which implies that 120593

120585(119909lowast) = 0 and 119909lowast isin 119878

120585 It follows

from Lemma 7 that 119909lowast solves VVI (1)Suppose that 119909lowast solves VVI (1) From Lemma 7 it follows

that there exists a vector 120585 isin 119861 such that 119909lowast isin 119878120585and so

120593120585(119909lowast) = 0 Since for all 120585 isin 119861 120593

120585(119909lowast) ge 0 we have 120595(119909lowast) =

min120585isin119861

120593120585(119909lowast) = 0

Thus (11) is a gap function for VVI (1) The last assertionis obvious from the definition of gap function

This completes the proof

Since 120595 is constructed based on the regularized gapfunction for VI

120585 we wish to call it a regularized gap function

forVVI (1)Theorem 8 indicates that in order to get a solutionof VVI (1) we only need to solve problem (24) In whatfollows wewill discuss some properties of the regularized gapfunction 120595

Theorem 9 If 119865119895(119895 = 1 119901) are continuously differen-

tiable then the regularized gap function 120595 is directionallydifferentiable in any direction 119889 isin R119899 and its directionalderivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (25)

where 119861(119909) = 120585 isin 119861 120593120585(119909) = min

120585isin119861120593120585(119909)

Proof It follows since the projection operator is nonexpan-sive that 120593

120585(119909) is continuous in (120585 119909)Thus 119861(119909) is nonempty

for any 119909 isin 119870 From Lemma 5 it follows that 120593120585(119909) is

continuously differentiable in 119909 and that

nabla119909120593120585 (119909) =

119901

sum

119895=1

120585119895119865119895 (119909) minus (

119901

sum

119895=1

120585119895nabla119909119865119895 (119909) minus 119866)(119867120585 (

119909) minus 119909)

(26)

is continuous in (120585 119909) It follows fromTheorem 1 of [23] that120595 is directionally differentiable in any direction 119889 isin R119899 andits directional derivative 1205951015840(119909 119889) is given by

1205951015840(119909 119889) = inf

120585isin119861(119909)

⟨nabla119909120593120585 (119909) 119889⟩ (27)

This completes the proof

By the directional differentiability of 120595 shown inTheorem 9 the first-order necessary condition of optimalityfor problem (24) can be stated as

1205951015840(119909 119910 minus 119909) ge 0 forall119910 isin 119870 (28)

If one wishes to solve VVI (1) via the optimization problem(24) we need to obtain its global optimal solution Howeversince the function 120595 is in general nonconvex it is difficult tofind a global optimal solution Fortunately we can prove thatany point satisfying the first-order condition of optimalitybecomes a solution of VVI (1) Thus existing optimizationalgorithms can be used to find a solution of VVI (1)

Theorem 10 Assume that 119865119895are continuously differentiable

and that the Jacobian matrixes nabla119909119865119895(119909) are positive definite for

all 119909 isin 119870 (119895 = 1 119901) If 119909lowast isin 119870 satisfies the first-ordercondition of optimality (28) then 119909lowast solves VVI (1)

Proof It follows fromTheorem 9 and (28) that for some 120585 isin119861(119909lowast)

⟨nablalowast

119909120593120585(119909lowast) 119910 minus 119909

lowast⟩ ge 0 (29)

holds for any 119910 isin 119870 This implies that 119909lowast is a stationary pointof problem (8) It follows from Lemma 6 that 119909lowast solves VI

120585

FromLemma 7 we see that119909lowast solves VVIThis completes theproof

From Theorems 8 and 10 it is easy to get the followingcorollary

Corollary 11 Assume that the conditions in Theorem 10 areall satisfied If 119909lowast isin 119870 satisfies the first-order condition ofoptimality (28) then 119909lowast is a global optimal solution of problem(24)

4 Stochastic Vector Variational Inequality

In this section we consider the stochastic vector variationalinequality (SVVI) First we present a deterministic refor-mulation for SVVI by employing the ERM method and theregularized gap function Second we solve this reformulationby the SAA method

In most important practical applications the functions119865119895(119895 = 1 119901) always involve some random factors or

uncertainties Let (ΩF 119875) be a probability space Takingthe randomness into account we get a stochastic vectorvariational inequality problem (SVVI) find a vector 119909lowast isin 119870such that

(⟨1198651(119909lowast 120596) 119910 minus 119909

lowast⟩ ⟨119865

119901(119909lowast 120596) 119910 minus 119909

lowast⟩)

119879

notin minus int R119901+ forall119910 isin 119870 as

(30)

where 119865119895 R119899 times Ω rarr R119899 are mappings and as is the

abbreviation for ldquoalmost surelyrdquo under the given probabilitymeasure 119875

Because of the random element 120596 we cannot generallyfind a vector 119909lowast isin 119870 such that (30) holds almost surelyThat is (30) is not well defined if we think of solving(30) before knowing the realization 120596 Therefore in orderto get a reasonable resolution an appropriate deterministicreformulation for SVVI becomes an important issue in thestudy of the considered problem In this section we willemploy the ERMmethod to solve (30)

Define120595 (119909 120596) = min

120585isin119861

120593120585 (119909 120596) (31)

where

120593120585 (119909 120596) = max

119910isin119870

⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119910⟩ minus

1

2

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2

119866

(32)

Journal of Applied Mathematics 5

The maximum in (32) is attained at

119867120585 (119909 120596) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909 120596)) (33)

120593120585 (119909 120596) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

(34)

The ERM reformulation is given as follows

min119909isin119870

Θ (119909) = E [120595 (119909 120596)] (35)

where E is the expectation operatorNote that the objective functionΘ(119909) contains the math-

ematical expectation In practical applications it is in generalvery difficult to calculate E[120595(119909 120596)] in a closed form Thuswe will have to approximate it through discretization Oneof the most popular discretization approaches is the sampleaverage approximation method In general for an integrablefunction 120601 Ω rarr R we approximate the expected valueE[120601(120596)] with the sample average (1119873

119896) sum120596119894isinΩ119896

120601(120596119894) where

1205961 120596

119873119896are independently and identically distributed

random samples of 120596 and Ω119896= 1205961 120596

119873119896 By the strong

law of large numbers we get the following lemma

Lemma 12 If 120601(120596) is integrable then

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120601 (120596119894) = E [120601 (120596)] (36)

holds with probability one

Let Θ119896(119909) = (1119873

119896) sum120596119894isinΩ119896

120595(119909 120596119894) Applying the above

technique we get the following approximation of (35)

min119909isin119870

Θ119896 (119909) (37)

In the rest of this section we focus on the case

119865119895 (119909 120596) = 119872119895 (

120596) 119909 + 119876119895 (120596) (38)

(119895 = 1 119901) where119872119895 Ω rarr R119899times119899 and 119876

119895 Ω rarr R119899 are

measurable functions such that

E [10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816

2

] lt +infin E [10038161003816100381610038161003816119876(120596)119895

10038161003816100381610038161003816

2

] lt +infin

119895 = 1 119901

(39)

This condition implies that

E[

[

119901

sum

119895=1

119872119895 (120596)]

]

2

lt +infin

E[

[

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin

(40)

and for any scalar 119888

E[

[

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816119888 +

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin (41)

The following results will be useful in the proof of theconvergence result

Lemma 13 Let 119891 119892 R119901 rarr R+

be continuous Ifmin120585isin119861

119891(120585) lt +infin andmin120585isin119861

119892(120585) lt +infin then

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816 (42)

Proof Without loss of generality we assume thatmin120585isin119861119891(120585) le min

120585isin119861119892(120585) Let 120585minimize 119891 and 120585minimize

119892 respectively Hence 119891(120585) le 119892(120585) and 119892(120585) le 119892(120585) Thus

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

= 119892 (120585) minus 119891 (120585)

le 119892 (120585) minus 119891 (120585)

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816

(43)

This completes the proof

Lemma 14 When 119865119895(119909 120596) = 119872

119895(120596)119909 +119876

119895(120596) (119895 = 1 119901)

the function 120593120585(119909 120596) is continuously differentiable in 119909 almost

surely and its gradient is given by

nabla119909120593120585 (119909 120596) =

119901

sum

119895=1

120585119895119865119895 (119909 120596)

minus (

119901

sum

119895=1

120585119895119872119895 (120596) minus 119866)(119867120585 (

119909 120596) minus 119909)

(44)

Proof Theproof is the same as that ofTheorem 32 in [21] sowe omit it here

Lemma 15 For any 119909 isin 119870 one has

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (45)

Proof For any fixed120596 isin Ω 120593120585(119909 120596) is the gap function of the

following scalar variational inequality find a vector 119909lowast isin 119870such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast 120596) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (46)

6 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

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

Journal of Applied Mathematics 5

The maximum in (32) is attained at

119867120585 (119909 120596) = Proj

119870119866(119909 minus 119866

minus1

119901

sum

119895=1

120585119895119865119895 (119909 120596)) (33)

120593120585 (119909 120596) = ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

minus

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

(34)

The ERM reformulation is given as follows

min119909isin119870

Θ (119909) = E [120595 (119909 120596)] (35)

where E is the expectation operatorNote that the objective functionΘ(119909) contains the math-

ematical expectation In practical applications it is in generalvery difficult to calculate E[120595(119909 120596)] in a closed form Thuswe will have to approximate it through discretization Oneof the most popular discretization approaches is the sampleaverage approximation method In general for an integrablefunction 120601 Ω rarr R we approximate the expected valueE[120601(120596)] with the sample average (1119873

119896) sum120596119894isinΩ119896

120601(120596119894) where

1205961 120596

119873119896are independently and identically distributed

random samples of 120596 and Ω119896= 1205961 120596

119873119896 By the strong

law of large numbers we get the following lemma

Lemma 12 If 120601(120596) is integrable then

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120601 (120596119894) = E [120601 (120596)] (36)

holds with probability one

Let Θ119896(119909) = (1119873

119896) sum120596119894isinΩ119896

120595(119909 120596119894) Applying the above

technique we get the following approximation of (35)

min119909isin119870

Θ119896 (119909) (37)

In the rest of this section we focus on the case

119865119895 (119909 120596) = 119872119895 (

120596) 119909 + 119876119895 (120596) (38)

(119895 = 1 119901) where119872119895 Ω rarr R119899times119899 and 119876

119895 Ω rarr R119899 are

measurable functions such that

E [10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816

2

] lt +infin E [10038161003816100381610038161003816119876(120596)119895

10038161003816100381610038161003816

2

] lt +infin

119895 = 1 119901

(39)

This condition implies that

E[

[

119901

sum

119895=1

119872119895 (120596)]

]

2

lt +infin

E[

[

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin

(40)

and for any scalar 119888

E[

[

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895 (120596)

10038161003816100381610038161003816119888 +

10038161003816100381610038161003816119876119895 (120596)

10038161003816100381610038161003816)]

]

lt +infin (41)

The following results will be useful in the proof of theconvergence result

Lemma 13 Let 119891 119892 R119901 rarr R+

be continuous Ifmin120585isin119861

119891(120585) lt +infin andmin120585isin119861

119892(120585) lt +infin then

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816 (42)

Proof Without loss of generality we assume thatmin120585isin119861119891(120585) le min

120585isin119861119892(120585) Let 120585minimize 119891 and 120585minimize

119892 respectively Hence 119891(120585) le 119892(120585) and 119892(120585) le 119892(120585) Thus

10038161003816100381610038161003816100381610038161003816

min120585isin119861

119891 (120585) minusmin120585isin119861

119892 (120585)

10038161003816100381610038161003816100381610038161003816

= 119892 (120585) minus 119891 (120585)

le 119892 (120585) minus 119891 (120585)

le max120585isin119861

1003816100381610038161003816119891 (120585) minus 119892 (120585)

1003816100381610038161003816

(43)

This completes the proof

Lemma 14 When 119865119895(119909 120596) = 119872

119895(120596)119909 +119876

119895(120596) (119895 = 1 119901)

the function 120593120585(119909 120596) is continuously differentiable in 119909 almost

surely and its gradient is given by

nabla119909120593120585 (119909 120596) =

119901

sum

119895=1

120585119895119865119895 (119909 120596)

minus (

119901

sum

119895=1

120585119895119872119895 (120596) minus 119866)(119867120585 (

119909 120596) minus 119909)

(44)

Proof Theproof is the same as that ofTheorem 32 in [21] sowe omit it here

Lemma 15 For any 119909 isin 119870 one has

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (45)

Proof For any fixed120596 isin Ω 120593120585(119909 120596) is the gap function of the

following scalar variational inequality find a vector 119909lowast isin 119870such that

⟨

119901

sum

119895=1

120585119895119865119895(119909lowast 120596) 119910 minus 119909

lowast⟩ ge 0 forall119910 isin 119870 (46)

6 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

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 Journal of Applied Mathematics

Hence 120593120585(119909 120596) ge 0 for all 119909 isin 119870 From (4) and (34) we have

1

2

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

2

119866

le ⟨

119901

sum

119895=1

120585119895119865119895 (119909 120596) 119909 minus 119867120585 (

119909 120596)⟩

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895 (119909 120596)

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816

le

1

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

(47)

and so

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866le

2

radic120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816 (48)

It follows from (4) that

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816le

1

radic120582min

10038161003816100381610038161003816119909 minus 119867

120585 (119909 120596)

10038161003816100381610038161003816119866

le

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119865119895 (119909 120596)

10038161003816100381610038161003816

(49)

This completes the proof

Now we obtain the convergence of optimal solutions ofproblem (37) in the following theorem

Theorem 16 Let 119909119896 be a sequence of optimal solutions ofproblem (37) Then any accumulation point of 119909119896 is anoptimal solution of problem (35)

Proof Let 119909lowast be an accumulation point of 119909119896 Without lossof generality we assume that 119909119896 itself converges to 119909lowast as 119896tends to infinity It is obvious that 119909lowast isin 119870 At first we willshow that

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) = E [120595 (119909

lowast 120596)] (50)

From Lemma 12 it suffices to show that

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (51)

It follows from Lemma 13 and the mean-value theorem that

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816120595 (119909119896 120596119894) minus 120595 (119909

lowast 120596119894)

10038161003816100381610038161003816

=

1

119873119896

sum

120596119894isinΩ119896

10038161003816100381610038161003816100381610038161003816

min120585isin119861

120593120585(119909119896 120596119894) minusmin120585isin119861

120593120585(119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816120593120585(119909119896 120596119894) minus 120593120585(119909lowast 120596119894)

10038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

(52)

where 119910119896119894 = 120582119896119894119909119896+ (1 minus 120582

119896119894)119909lowast with 120582

119896119894isin [0 1] Because

lim119896rarr+infin

119909119896= 119909lowast there exists a constant119862 such that |119909119896| le 119862

for each 119896 By the definition of 119910119896119894 we have |119910119896119894| le 119862 Hencefor any 120585 isin 119861 and 120596

119894isin Ω119896

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119901

sum

119895=1

120585119895119865119895(119910119896119894 120596119894)

minus (

119901

sum

119895=1

120585119895nabla119909119865119895(119910119896119894 120596119894) minus 119866)

times (119867120585(119910119896119894 120596119894) minus 119910119896119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+ (

119901

sum

119895=1

10038161003816100381610038161003816120585119895

10038161003816100381610038161003816

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

10038161003816100381610038161003816119867120585(119910119896119894 120596119894) minus 11991011989611989410038161003816100381610038161003816

le

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816nabla119909119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816+ |119866|)

times

119901

sum

119895=1

10038161003816100381610038161003816119865119895(119910119896119894 120596119894)

10038161003816100381610038161003816

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

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

Journal of Applied Mathematics 7

le (

2

120582min

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816+

2

120582min|119866| + 1)

times

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

=

2119862

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816) (

2

120582min|119866| + 1)

+

2

120582min(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)(

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(53)

where the second inequality is from Lemma 15 Thus10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

1

119873119896

sum

120596119894isinΩ119896

max120585isin119861

10038161003816100381610038161003816nabla119909120593120585(119910119896119894 120596119894)

10038161003816100381610038161003816

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

le

2119862

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

2

+

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

119901

sum

119895=1

(

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816119862 +

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

times (

2

120582min|119866| + 1)

+

2

120582min

10038161003816100381610038161003816119909119896minus 119909lowast10038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

(

119901

sum

119895=1

10038161003816100381610038161003816119872119895(120596119894)

10038161003816100381610038161003816)

times (

119901

sum

119895=1

10038161003816100381610038161003816119876119895(120596119894)

10038161003816100381610038161003816)

(54)

From (40) and (41) each term in the last inequality aboveconverges to zero and so

lim119896rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) minus

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894)

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (55)

Since

lim119896rarrinfin

1

119873119896

sum

120596119894isinΩ119896

120595 (119909lowast 120596119894) = E [120595 (119909

lowast 120596)] (56)

(50) is trueNow we are in the position to show that 119909lowast is a solution

of problem (35)

Since 119909119896 solves problem (37) for each 119896 we have that forany 119909 isin 119870

1

119873119896

sum

120596119894isinΩ119896

120595 (119909119896 120596119894) le

1

119873119896

sum

120596119894isinΩ119896

120595 (119909 120596119894) (57)

Letting 119896 rarr infin above we get from Lemma 12 and (50) that

E [120595 (119909lowast 120596)] le E [120595 (119909 120596)] (58)

which means that 119909lowast is an optimal solution of problem (35)This completes the proof

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11171237 11101069 and 71101099) bythe Construction Foundation of Southwest University forNationalities for the subject of applied economics (2011XWD-S0202) and by Sichuan University (SKQY201330)

References

[1] F Giannessi Ed Vector Variational Inequalities and VectorEquilibria vol 38 Kluwer Academic Dordrecht The Nether-lands 2000

[2] S J Li and C R Chen ldquoStability of weak vector variationalinequalityrdquoNonlinear AnalysisTheory Methods ampApplicationsvol 70 no 4 pp 1528ndash1535 2009

[3] Y H Cheng and D L Zhu ldquoGlobal stability results for the weakvector variational inequalityrdquo Journal of Global Optimizationvol 32 no 4 pp 543ndash550 2005

[4] X Q Yang and J C Yao ldquoGap functions and existence ofsolutions to set-valued vector variational inequalitiesrdquo Journalof OptimizationTheory and Applications vol 115 no 2 pp 407ndash417 2002

[5] N-J Huang and Y-P Fang ldquoOn vector variational inequalitiesin reflexive Banach spacesrdquo Journal of Global Optimization vol32 no 4 pp 495ndash505 2005

[6] S J Li H Yan and G Y Chen ldquoDifferential and sensitivityproperties of gap functions for vector variational inequalitiesrdquoMathematical Methods of Operations Research vol 57 no 3 pp377ndash391 2003

[7] K W Meng and S J Li ldquoDifferential and sensitivity propertiesof gap functions for Minty vector variational inequalitiesrdquoJournal of Mathematical Analysis and Applications vol 337 no1 pp 386ndash398 2008

[8] N J Huang J Li and J C Yao ldquoGap functions and existence ofsolutions for a system of vector equilibrium problemsrdquo Journalof OptimizationTheory and Applications vol 133 no 2 pp 201ndash212 2007

[9] G-Y Chen C-J Goh and X Q Yang ldquoOn gap functions forvector variational inequalitiesrdquo inVectorVariational Inequalitiesand Vector Equilibria vol 38 pp 55ndash72 Kluwer AcademicDordrecht The Netherlands 2000

[10] R P Agdeppa N Yamashita and M Fukushima ldquoCon-vex expected residual models for stochastic affine variationalinequality problems and its application to the traffic equilibriumproblemrdquo Pacific Journal of Optimization vol 6 no 1 pp 3ndash192010

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

[11] X Chen and M Fukushima ldquoExpected residual minimizationmethod for stochastic linear complementarity problemsrdquoMath-ematics of Operations Research vol 30 no 4 pp 1022ndash10382005

[12] X Chen C Zhang and M Fukushima ldquoRobust solution ofmonotone stochastic linear complementarity problemsrdquoMath-ematical Programming vol 117 no 1-2 pp 51ndash80 2009

[13] H Fang X Chen andM Fukushima ldquoStochastic1198770matrix lin-

ear complementarity problemsrdquo SIAM Journal onOptimizationvol 18 no 2 pp 482ndash506 2007

[14] H Jiang and H Xu ldquoStochastic approximation approaches tothe stochastic variational inequality problemrdquo IEEE Transac-tions on Automatic Control vol 53 no 6 pp 1462ndash1475 2008

[15] G-H Lin andM Fukushima ldquoNew reformulations for stochas-tic nonlinear complementarity problemsrdquo Optimization Meth-ods amp Software vol 21 no 4 pp 551ndash564 2006

[16] C Ling L Qi G Zhou and L Caccetta ldquoThe 1198781198621 property ofan expected residual function arising from stochastic comple-mentarity problemsrdquoOperations Research Letters vol 36 no 4pp 456ndash460 2008

[17] M-J Luo and G-H Lin ldquoStochastic variational inequalityproblems with additional constraints and their applications insupply chain network equilibriardquo Pacific Journal of Optimiza-tion vol 7 no 2 pp 263ndash279 2011

[18] M J Luo and G H Lin ldquoExpected residual minimizationmethod for stochastic variational inequality problemsrdquo Journalof OptimizationTheory and Applications vol 140 no 1 pp 103ndash116 2009

[19] M J Luo and G H Lin ldquoConvergence results of the ERMmethod for nonlinear stochastic variational inequality prob-lemsrdquo Journal of OptimizationTheory and Applications vol 142no 3 pp 569ndash581 2009

[20] C Zhang and X Chen ldquoStochastic nonlinear complementarityproblem and applications to traffic equilibrium under uncer-taintyrdquo Journal of OptimizationTheory andApplications vol 137no 2 pp 277ndash295 2008

[21] M Fukushima ldquoEquivalent differentiable optimization prob-lems and descent methods for asymmetric variational inequal-ity problemsrdquoMathematical Programming vol 53 no 1 pp 99ndash110 1992

[22] R T Rockafellar Convex Analysis Princeton University PressPrinceton NJ USA 1970

[23] W Hogan ldquoDirectional derivatives for extremal-value func-tions with applications to the completely convex caserdquo Opera-tions Research vol 21 pp 188ndash209 1973

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