Existence and Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

Abstract

An existence result for the solution set of symmetric vector quasi-equilibrium problems that allows for discontinuities is obtained. Moreover, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are established.

1. Introduction

Well-posedness of optimization problems was first studied by Tykhonov [1] in 1966. Since then, the notion of well-posedness has been extended to different kinds of optimization problems (see [2–5]). In the book edited by Lucchetti and Revalski [6], Loridan gave a survey on some theoretical results of well-posedness, approximate solutions, and variational principles in vector optimization. Well-posedness for constrained optimization problems was first studied by Levitin and Polyak [7]. The study of Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints comes from [8]. Recently, this research was extended to general constrained vector optimization problems [9], generalized variational inequality problems with functional constraints [10], and vector equilibrium problems with functional constraints [11].

In 2003, Fu [12] introduced the symmetric vector quasi-equilibrium problem (for short ) and studied existence conditions of . is a generalization of the equilibrium problem, proposed by Blum and Oettli [13], and a unified model of several problems, for example, vector optimization problems, problems of vector Nash equilibria, vector variational inequalities, and vector complementarity problems. Farajzadeh [14] considered existence theorem of the solution of symmetric vector quasi-equilibrium problems in the Hausdorff topological vector space and answered the open question raised by Fu [12]. In [15], Li et al. obtained existence results for two classes of generalized vector quasi-equilibrium problems. Zhang [16] introduced generalized Levitin-Polyak well-posedness for and obtained sufficient conditions for the generalized Levitin-Polyak well-posedness of .

In this paper, we will introduce existence and well-posed theorem of for discontinuous vector-valued mapping, which extend the corresponding result in [12] in metric space. Then, by using the conditions of the existence theorem of the solutions to (SVQEP) in [14], we obtain sufficient conditions for the generalized Levitin-Polyak well-posedness of , which improve the result of [16, Theorem 4.1].

The paper is organized as follows. In Section 2, we present some preliminary concepts. In Section 3, we prove the existence theorem of that allows for discontinuities. In Section 4, we obtain generalized Levitin-Polyak well-posed results for .

2. Preliminaries

Let and be real locally convex Hausdorff spaces, and let and be nonempty subsets of and , respectively. Let be a real Hausdorff topological vector space and a closed convex and pointed cone with .

Assume that is a nonempty subset. A point is called a minimal point of , if . A point is called a weak minimal point of if . A point is called an -weak minimal point of if . By , , and we denote the sets of all minimal points, weak minimal points, and -weak minimal points of , respectively. Obviously, .

Let be a nonempty subset of and let be a vector-valued mapping. Consider the following vector-valued optimization problem: A point is called a (weak) minimizer of if is a minimal point (weak minimal point) of the set ; namely, for every , The set of all (weak) minimizers of is denoted by (). Obviously, .

Let and be two set-valued mappings and let be two vector-valued mappings. Fu [12] defined a class of symmetric vector quasi-equilibrium problems (for short ), which consist in finding such that , , and We call a solution of and denote by the solution set of . The equilibrium problem contains optimization problem as special case (see [17, 18]). The problem is a generalization of quasi-optimization problem proposed by Crespi and Tan [19].

Definition 1. Let be an ordered topological vector space, let be a nonempty convex subset of a vector space , and let be a vector mapping. (i) is called -convex if, for every and for every , one has (ii) is called proper -quasiconvex if, for every and , one has either or .(iii) is said to be natural -quasiconvex on if, for every , , there exists such that (iv) is called strict -convex if, for every with and for every , one has

Remark 2. It is clear that every strict -convex mapping is convex, and every convex or proper -quasiconvex mapping is natural -quasiconvex. A vector mapping may be -convex and not proper -quasiconvex and conversely (see [20]). A vector mapping may be natural -quasiconvex but neither -convex nor proper -quasiconvex.

Let be a set-valued map. is said to be upper semicontinuous (. for short) at if, for any open set , there exists a neighborhood of such that If is . at each point of , then is said to be .    is said to be lower semicontinuous (. for short) at if, for any and any neighborhood of , there exists a neighborhood of such that ; we have is said to be . if is . at every point of . Moreover, is said to be continuous if is both . and . From [21], we can see that is lower semicontinuous at if and only if, for any and any net with , there is a net such that and . is said to be closed if the graph of , that is, , is a closed set in .

Let be a set-valued map. A point is called a fixed point of the set-valued map if .

Lemma 3. Let be a metric space, let , and let be a sequence converging to in . Let and be two elements in such that and assume that there exist two sequences and valued in such that where and . Then for all sufficiently large one has .

The proof of Lemma 3 is similar to that of Lemma  1 in [22]; for details, see [22].

3. Existence Results for (SVQEP)

Throughout this section, let , , and be metric spaces, let the sets and be nonempty, convex, and compact subsets, and let be a closed, convex pointed cone with .

Theorem 4. Assume that (1) and are continuous, and for each , , are nonempty, closed convex subsets;(2)for any , for all , and converging to , the following conditions hold: (3)for any and converging to such that , there exists such that ; for any and converging to such that , there exists such that ;(4) and are compact sets;(5)for any fixed , is proper -quasiconvex in ; for any fixed , is proper -quasiconvex in .
Then (SVQEP) has a solution.

Proof. Let us define set-valued maps and by Similar to the proof of [12, Section 2 Theorem], the set is nonempty and convex. We only need to show the following.
(I) For all , the set is closed. In fact, let a sequence and ; we need to show that . It follows from and the closedness of that . Since we have If there exists such that from condition , ; thus, From Lemma 3, for large enough. It is a contradiction.
(II) is u.s.c., since is a compact set and . By [12, Lemma ], we need only to show that the set-valued map is closed. Let a sequence , and . Since and the set-valued map is continuous, we have . For any , since is l.s.c., there is a sequence , such that . Since , we get
If there exists such that , from (ii), From (iii), there exists such that By Lemma 3, , which contradicts (16).

Remark 5. It is clear that if and are continuous mappings and condition holds, then conditions , , and of Theorem 4 hold. The following example shows that Theorem 4 improves [12, Section 2. Theorem].

Example 6. Suppose that , , and and let and be defined as and , respectively. For all , let It is clear that the mappings and are not continuous, but all the conditions of Theorem 4 hold.
Moreover, let ; we can get from [23, Lemma 2.2] that the mappings and are natural -quasiconvex but not demicontinuous (see [14, Definition 2.4]). Therefore, Theorem 4 is different from [14, Theorem 3.1].

4. Well-Posedness of (SVQEP)

In this section, we discuss the notion of generalized Levitin-Polyak well-posedness for .

Definition 7. A sequence is called a Levitin-Polyak approximating solution sequence (in short LP sequence) for if there exists with such that

Definition 8. The problem is said to be generalized Levitin-Polyak well-posed (in short LP well-posed) if (i);(ii)for every LP sequence , there exist a subsequence and an element such that .

Let us illustrate the notion of generalized LP well-posedness by some examples.

Example 9. Let , , and . For all , let and . Set-valued mappings and are defined by ; is generalized LP well-posed.

Example 10. Let , , and . For all , let and . Set-valued mappings and are defined by and ; is not generalized LP well-posed.

Remark 11. (i) Generalized LP well-posedness of implies that the set is compact.
(ii) It is easy to see that the notion of well-posedness of generalizes the notion of generalized LP well-posedness of vector equilibrium problem introduced in [20].

Theorem 12. Let with . Under the assumptions of Theorem 4, is generalized LP well-posed.

Proof. Let with and Since are continuous and compact-valued, there exist a subsequence and an element such that . If there exists , such that , from Theorem 4(ii), From Theorem 4(iii), there exists such that By Lemma 3, when is large enough, which is a contradiction. Therefore, Similarly, From Definition 7, is generalized LP well-posed.

Definition 13 (see [23]). Let be a topological space, and let be a topological vector space. A function is said to be demicontinuous if is closed in for each closed half space .

Lemma 14 (see [23]). Let be a topological space, a topological vector space, and a demicontinuous function. Then for any , the composite function is continuous, where is the topological dual space of .

Let be a closed convex and pointed cone with and By [24, p. 165, Theorem 2], there exists such that We get that .

Lemma 15. If for all , then .

Proof. If we assume that there exists such that for all , then we have If not, there exist , , and such that . Thus, It is a contradiction. Thus, (30) holds. By [24, p. 165, Theorem 2], there exists such that, for all , and for all , Then, and . This, however, contradicts the fact that for all .

Theorem 16. Assume that (1) and are continuous, and for each , , are nonempty, closed convex subsets;(2) are demicontinuous;(3)for any fixed , is natural -quasiconvex in ; for any fixed , is natural -quasiconvex in .
Then is generalized LP well-posed.

Proof. From [14, Theorem 3.1], . Let with and Since are continuous and compact-valued, there exist a subsequence and an element such that . If there exists , such that
for all , we have Then, Since is l.s.c. at , there exists such that . By Lemma 14, when is large enough. By Lemma 15, which contradicts (34). Therefore, is generalized LP well-posed.

Let be a closed set, let , and consider a cone to be closed, convex, pointed, and with nonempty interior. We consider the following vector optimization problem.Find such that

As a consequence of Theorem 16, we have the following well-posed result for the solution of .

Corollary 17. Let be a compact set, and let be demicontinuous and natural -quasiconvex on . Then, is generalized LP well-posed.

Remark 18. (i) From Remark 2, every strict -convex mapping is natural -quasiconvex. Then Theorem 16 improves [16, Theorem 4.1].
(ii) From [23, Lemma ], every natural -quasiconvex function is -quasiconvex. In the assumption of being continuous, Corollary 17 is a special case of [18, Theorem 4.2].

Conflict of Interests

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

Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable comments and suggestions, which helped in improving the paper. This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401), by Humanities and Social Science Project of Ministry of Education of China (14XJC), and by Soft Science Research Project of Sichuan Province (2014ZR0027).