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Abstract and Applied Analysis
Volume 2014, Article ID 948270, 8 pages
http://dx.doi.org/10.1155/2014/948270
Research Article

Generalized Stampacchia Vector Variational-Like Inequalities and Vector Optimization Problems Involving Set-Valued Maps

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

Received 22 June 2014; Accepted 10 August 2014; Published 27 August 2014

Academic Editor: S. D. Purohit

Copyright © 2014 Yanfei Chai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first obtain that subdifferentials of set-valued mapping from finite-dimensional spaces to finite-dimensional possess certain relaxed compactness. Then using this weak compactness, we establish gap functions for generalized Stampacchia vector variational-like inequalities which are defined by means of subdifferentials. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex set-valued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized Stampacchia vector variational-like inequality via a Fan-KKM lemma.

1. Introduction

Vector variational inequality (VVI) was first introduced by Giannessi [1] in finite-dimensional spaces. Since then, much attention has been given to investigate VVIs and their generalizations. For details on VVI and their generalizations, we refer to [24] and the references therein. The vector variational-like inequality (VVLI), a generalization of VVI, was studied in [59] with applications in vector optimization problem (VOP). In [7], the equivalences among Minty VVLI, Stampacchia VVLI, both for -subdifferentiable functions (see [10]) and for nondifferentiable nonconvex VOP were established and an existence theorem for the so-called generalized weakly efficient solutions of nondifferentiable nonconvex VOP was obtained through the relationship between VVLI and VOP. The VVLI approach was also used in [9, 11] to prove some existence theorems of generalized efficient solutions for nondifferentiable invex VOP. Particularly, VVLI with set-valued mapping was also considered in [1214].

Note that some results regarding gap functions for VVLI with set-valued mappings have appeared in the literature [12, 14, 15], but so far to the best of our knowledge there is no result available in the literature about a gap function for a generalized Stampacchia vector variational-like inequality (GVVLI) which is defined by means of generalized subdifferentials of set-valued mapping (see Definition 3). In this paper, we introduce gap functions for our GVVLI.

Note also that there are some papers discussing solution relationships between set-valued optimization problems and vector variational-like inequalities. Miholca [13] and Zeng and Li [16] considered several kinds of generalized invexity for set-valued mappings and established some solution relationships between set-valued optimization problems and generalized vector variational-like inequalities problems (() and () in [13, 16], resp.), but they did not consider the existence results for their solutions.

Inspired and motivated by the works [7, 1216], in this paper, we study GVVLI. We first establish generalized subdi erential mapping of set-valued mapping from finite-dimensional space into finite-dimension is an asymptotically compact, closed and convex-valued mapping, and then, we introduce gap functions for our GVVLI problems. In the final section of this paper, we discuss solution relationships between GVVLI and vector optimization problem involving set-valued mapping (VOP); then we obtain existence of solutions for GVVLI by using a Fan-KKM lemma and consequently an existence result for weak efficient solution of VOP is established.

2. Preliminaries and Notations

Let be the dimensional space, where is a given positive integer. Set where the symbol denotes the transpose. A nonempty subset of is said to be a cone if for all ; is said to be a convex cone if is a cone and ; is a closed cone if is a cone and closed; and is called a pointed cone if is a cone and . Clearly, is a closed, convex, and pointed cone of , and int  is a convex pointed cone of . We consider the orderings induced by and int  in the following form.

Let :

In the following sections, we denote . Let be the -dimensional space. Denote by the space of all the continuous linear mappings from to and by the value of at . Let . For , denote by the distance from to ; that is, . Let be a set-valued mapping. The graph, the epigraph, and the domain of are defined, respectively, by

Definition 1 (see [17, 18]). Let ; the contingent cone of at is defined by

Definition 2 (see [17, 18]). Let and let a pair be given. The contingent epiderivative of at is the single-valued mapping from to defined by

is called epidifferentiable at if the contingent epiderivative of at exists.

Recall that the contingent cone consists of all tangent vectors with and for all and .

Definition 3 (see [19]). Suppose that is epidifferentiable at ; then(i)a continuous linear map , with , for all is called a subgradient of at (ii)the set ,   of all subgradients of at is called the subdifferential of at .

The following notions are based on the concept of contingent epiderivative.

Lemma 4 (see [19]). Let and be real normed spaces, a subset of , and a convex cone. , , and let contingent epiderivative of at exist. Then the subdifferential is convex.

Lemma 5 (see [19]). Let assumptions of Lemma 4 be satisfied and let be closed. If all subgradients are bounded, then the subdifferential is closed in the linear space of all bounded maps.

For the spacial case, where and , linear maps are always bounded, so the subdifferential is closed whenever is closed.

Corollary 6. If , , and , then subdifferential of at is a closed and convex set in the linear space .

We propose the following relaxed compactness which will be needed for the following sections.

Definition 7 (see [20]). (i) Let and be in . The sequence is said to pointwisely converge to and written as if for all .
(ii) Let and be in . The sequence is said to asymptotically pointwisely converge to if(a)sequence is bounded and it has a subsequence and such that ;(b)sequence with and the sequence has a subsequence which pointwisely converges to some .
(iii) A subset is called asymptotically pointwisely compact or asymptotically -compact if each sequence has a subsequence which asymptotically pointwisely converges to .
(iv) If, in (iii), pointwise convergence, that is, -lim, is replaced by convergence, that is, lim, a subset is called asymptotically compact.

Remark 8. (i) If and are finite dimensional, a convergence occurs if and only if the corresponding pointwise convergence does.
(ii) If and are finite dimensional, every subset is asymptotically -compact and asymptotically compact.
(iii) If and are finite dimensional, then every subset of is asymptotically -compact.

Remark 9. By Remark 8, if assumptions of Corollary 6 are satisfied and exists at , then is an asymptotically compact closed and convex subset in .

Remark 10. From Remark 9, if for all , whenever exists at , then each sequence has a subsequence which asymptotically converges to some . In the following sections we always assume that for all , whenever exists at .

Example 11. Let , , and and let be defined by Then and for all ; therefore, . Obviously, for all .

Definition 12. A set is said to be an invex set if there exists a function such that , , and .

Remark 13. Obviously, the convex set is a particular case of the invex set if , but the converse does not hold; see the following example.

Example 14. Let and let be defined by We note that is an invex set with respect to the above .

Throughout this paper, we always assume that is an invex subset of , the function is defined on , that is, , and is a set-valued mapping with existing at every and . By Remark 9, we obtain , when it exists, a set-valued mapping with an asymptotically compact closed convex-valued at . Furthermore, under mild assumptions, has certain compactness (see Remark 10). Next, we consider the following strong and weak generalized Stampacchia vector variational-like inequalities (GVVLI):(SGVVLI) find and such that and(WGVVLI) find such that and there exist satisfying We denote by and the solution sets of and , respectively. Obviously, .

3. Gap Functions

In this section, we assume that defined on is an open map such that for all . Then, we introduce the concept of gap functions for our SGVVLI and WGVVLI.

Definition 15. Let be the domain of SGVVLI (resp., WGVVLI). A function is said to be a gap function for SGVVLI (resp., WGVVLI) if it satisfies the following properties:(i), with ;(ii) if and only if (where ) solves SGVVLI (resp., WGVVLI).

Let with , , and . Denote that is, is the th component of , . Now, we define as where such that exists. For and , let where   is the set of all operators from to . Let , , and ; then , . We also define as follows:

Lemma 16. Let be a normed space and an asymptotically compact closed convex subset; then there exists such that .

Proof. Let ; then for any , there exists such that So sequence is bounded; by assumption being an asymptotically compact closed convex subset, there exists a subsequence of such that . For , by applying Hahn-Banach theory, there exists such that , . Hence, on the one hand, on the other hand, Therefore, .

Theorem 17. Suppose that for all with such that for all . If the function is an open map such that for all . Then, the following statements are true.(i)The function defined by (11) is a gap function for SGVVLI.(ii)The function defined by (13) is a gap function for WGVVLI.

Proof. (i) By assumptions, for each with , is an asymptotically compact closed convex set. Then, the function given by (11) is well defined and from Lemma 16 we have It is immediate that so that , with . Let with . We note that if and only if there exists such that or which is equivalent to that is, .
(ii) Since for any given with and we obtain We assume that solves WGVVLI. Then, for any , there is a such that which implies that Thus an operator from to has been defined. It follows that and Therefore, and so Since for all with , it follows that . Conversely, suppose that and . Let Since , for with , there exist such that or equivalently which implies Observe that for all . Since is asymptotically compact closed convex set and by Remark 10, for any , has an asymptotically convergent subnet with limit . Without loss of generality, we may assume that asymptotically converges to . Consequently, an operator has been defined; that is, and or which imply or Taking the limit for in (31) or (32), we obtain Then, for any , there exists such that which implies that, for any , there exists such that that is, . This completes the proof.

Example 18. Let , , and and let and be defined by Then, for all . Since and therefore, we get when , and, consequently, So, hold for all .

4. Existence Result for Vector Optimization Problems

Vector variational inequalities (or their generalized form) have been shown to be a useful tool in vector optimization. Some authors have proved the equivalence between them; see [8, 13, 16]. In this section, we prove the existence of generalized weakly efficient solutions for the following invex vector optimization problem (VOP) through the relationship between GVVLI and VOP by using a Fan-KKM lemma: where is a set-valued mapping as above and is an invex set.

We denote set-valued optimization problems (44) as (SOP). Let , .

Definition 19. A pair is called weak efficient solution of on if .

The set of all weak efficient solutions of (SOP) is denoted by .

Definition 20. Let be a set-valued mapping, , and . is called strict pseudoinvex with respect to at if , , and , .

Remark 21. We note that of Example 18 is strict pseudoinvex with respect to at . In fact, since for all and taking any , we have and obviously , where and .

Definition 22. Let be a set-valued mapping. is said to be with respect to on if , , .

Definition 23. is said to be upper semicontinuous at if for any neighbourhood of the origin in there exists a neighbourhood of the origin in such that

Lemma 24 (Lemma 1.1, [21]). Let a multifunction . If is compact-valued multifunction, then is upper semicontinuous if and only if for every net in that satisfies for some the net has a subnet converging to a point in .

The following lemma will give a similar but more generalized result.

Lemma 25. Let a multifunction . If is asymptotically closed compact-valued multifunction and upper semicontinuous such that for all and , then for every net in that satisfies for some the net has a subnet asymptotically converging to a point in .

Proof. Take a system , of neighbourhoods of the origin in such that , where stands for , , and and is an ordered index set. From the upper semicontinuity of and , one claims that for any there exists such that for all . Consequently, can be written as with and . From the asymptotic compactness of and , without loss of generality, we may assume that asymptotically converges to . Furthermore, since for all and the closedness assumption of , we can conclude that . This implies that asymptotically converges to . The proof is complete.

Proposition 26. Let be an invex set with respect to and let be with respect to the same on . Let , , and . Then , , , and

Proof. Taking arbitrary elements and , for all , we define a sequence with Since is an invex set and is a map, it follows for all So, is a sequence in the epigraph of converging to . Moreover we obtain Consequently, we get implying that It is clear that By the definition of the subdifferential , for all , we have that is, Considering (51) and (54), we conclude that .

Proposition 27. Let be a nonempty invex subset of and let be an open mapping such that it is affine in the first argument and , . Let be strict pseudoinvex with respect to the same and for all with such that for all . If and is upper semicontinuous at , then .

Proof. Let . Consider any and any sequence with . Since is invex, Since , we have That is, Since exists and , there exists such that as . Moreover, at every with and is strict pseudoinvex with respect to , so for all we have Since is affine in the first argument and , , we have that is, where is a closed cone. By assumptions and Remark 9, we get that is asymptotically compact closed convex set in . Furthermore, being upper semicontinuous at , and by Remark 10 and Lemma 25, consequently, asymptotically converges to some with . If sequence is bounded, without loss of generality, we may assume that ; otherwise, if sequence with , we may also assume sequence ; from these facts and being a closed cone, it follows that Thus, , such that Hence, .

Remark 28. Let and ; then is an invex set and is affine in the first argument and continuous in the second argument, where .

Proposition 29. Let be a nonempty invex set of and subdifferentiable and with respect to the same . If is a solution of , then it is also a weakly efficient solution of VOP.

Proof. Suppose that is a solution of WGVVLI, but not a generalized weakly efficient solution of VOP. Then there exist such that . That is, such that . Since is subdifferentiable and with respect to the same , by Proposition 26, for each , we have , which contradicts .

Let be a nonempty convex subset of a vector space . A mapping is said to be a mapping if for each nonempty finite subset of , , where denotes the convex hull of , and .

The following form of lemma appeared in [22].

Lemma 30. Let be a convex subset of a Hausdorff topological vector space , . Let be set-valued maps such that the following conditions hold: is a mapping on ;for all , is closed;if there is a nonempty subset of such that the intersection is compact and is contained in a compact convex subset of ,then

Theorem 31. (i) Let be a nonempty convex subset of and let be an open mapping such that it is affine in the first argument and continuous in the second argument and , ,  . Let be with respect to such that exists and is upper semicontinuous at every with and that for all . If is upper semicontinuous at each with .
(ii) Assume that there exists a nonempty compact convex set such that for each , , such that , where .
Then, VOP has a generalized weakly efficient solution.

Proof. We define a multivalued map by where such that . Then, , is closed in . Indeed, let be a sequence in such that . Let such that exists. is epidifferentiable at , so for each there exists such that . Furthermore, implies there exist such that Since is upper semicontinuous at , and considering also Remark 10, possesses an asymptotically convergent subnet. Without loss of generality, we may assume that the (or ) and . Since is continuous in the second variable and is a closed set, it is easy to check that . So, is a closed set. Next, we claim that is a mapping on . Suppose that is not a mapping; then there exist and with such that . Thus, such that, for any , Since is a convex cone, we have Since is affine, we get But from , , we have . Thus , which contradicts the fact , so is a mapping. By condition (ii), is a closed subset of a compact set and hence compact. Then by Lemma 30, . That is, , such that and , . Thus, . From Proposition 29, we obtain . The proof is complete.

Remark 32. We note that our assumptions in Theorem 31 are totally different from those of Theorem  8 in [5] and Theorem  4.5 in [9] since we assume that is upper semicontinuous and for all while correspondences and in [5] and [9], respectively, were supposed to be pseudomonotone and needs to be closed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Science Foundation of China (nos. 61373174 and 11301407).

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