Optimality Conditions for ε-Vector Equilibrium Problems

Abstract

By virtue of the separation theorem of convex sets, a necessary condition and a sufficient condition for ε-vector equilibrium problem with constraints are obtained. Then, by using the Gerstewitz nonconvex separation functional, a necessary and sufficient condition for ε-vector equilibrium problem without constraints is obtained.

1. Introduction

As the unified model of vector optimization problems, vector variational inequality problems, variational inclusion problems, and vector complementarity problems, vector equilibrium problems have been intensively studied. The existence results for various kinds of vector equilibrium problems have been established, for example, see [1–5] and the references therein. But so far, there are few papers which deal with the properties of the solutions for vector equilibrium problems. Giannessi [3] obtained sufficient conditions for efficient solution and weakly efficient solution to the vector variational inequalities in finite dimensional spaces. Gong [6] obtained some optimality conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to vector equilibrium problems with constraints by using the separation theorem of convex sets. Gong [7] the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to some vector equilibrium problems without constraints.

On the other hand, in some situations, it may not be possible to find an exact solution for an optimization problem, or such an exact solution simply does not exist, for example, if the feasible set is not compact. Thus, it is meaningful to look for an approximate solution instead. There are also many papers to investigate the approximate solution problem, such as [1, 8–11]. Kimura et al. [1] obtained several existence results for -vector equilibrium problem and the lower semicontinuity of the solution mapping of -vector equilibrium problem. Anh and Khanh [10] have considered two kinds of solution sets to parametric generalized -vector quasiequilibrium problems and established the sufficient conditions for the Hausdorff semicontinuity (or Berge semicontinuity) of these solution mappings. Gupta and Mehra [9] introduced two new concepts of approximate saddle points and investigated two types of approximate solutions for a vector optimization problem in Banach space setting. X. B. Li and S. J. Li [11] obtained the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.

The aim of this paper is to characterize optimality conditions for -vector equilibrium problems. The paper is organized as follows. In Section 2, we recall the main notions and definitions. In Section 3, we obtain some optimality conditions for -vector equilibrium problems and -vector equilibrium problems with constraints, respectively.

2. Preliminaries

Let and be two real Hausdorff topological vector spaces and be a real locally convex Hausdorff topological vector space. Assume that and are two pointed closed convex cones in and with nonempty interior and , respectively. Let and be the topological dual space of and . Denote the dual cone of by and by : Letting be given, we have that is a weak* compact base of .

Definition 1. Let be a nonempty convex subset of , and let be a vector-valued mapping. is said to be -convex if and only if, for all and ,

Definition 2 (see [12]). Given , the Gerstewitz nonconvex separation function is defined by
Next, we give some useful properties of the above scalarization functions.

Lemma 3 (see [13]). Let . The following properties hold: (i);(ii);(iii) is a continuous function;(iv) is strictly monotonically increasing, that is, if .

3. Optimality Conditions

In this section, we first deal with the following -vector equilibrium problem with constraints (for short -VEPC): find such that and the constraint set where is a nonempty subset of , is a vector-valued mapping, is a vector-valued mapping, , and is a positive real number.

If , , and if is a solution of -VEP, then is a solution of -efficient solution of vector optimization problem of , where is a vector-valued mapping.

First, we give some necessary and sufficient conditions for -vector equilibrium problem with constraints by using the separation theorem of convex sets.Convexity Assumption: for all and , there exists such that

Remark 4. (i) Assumption () does not require that be a convex set.
(ii) We say that is -convex-like in if satisfies (6) and that is -convex-like if satisfies (7).
(iii) If is a convex set, is -convex in , and is -convex, then Assumption () is satisfied.

Theorem 5. Let and be a nonempty subset of . Let be a vector-valued mapping with , for all and let   be a vector-valued mapping. Assume that () is satisfied and there exists   such that . If is a solution of -VEPC, then there exists and such that where is a positive real number.

Proof. Let be a solution of -VEPC. We consider the set By assumptions, it is clear that and is an open set. Now, we show that . If not, by the definition of , then there exists such that Thus, . This contradicts that is a solution of -VEPC. Hence, .
Next, we show that is a convex set. Let and . By the definition of , there exist such that Then, by (11), we have By assumptions, there exists satisfying and . Namely, is a convex set.
Thus, by the separation theorem of convex sets, there exists such that Let , there exists such that and . Hence, for every , , and , we have and . By (13), Letting , we get , for all . Since is closed convex cone, by the continuity of , , for all ; that is, . Similarly, . We next show . In fact, if , by (13), we have By assumptions, there exists such that . Then, we get . For the above , we have By (15), . This is a contradiction. Namely, .
It is clear that , for all , and . Thus, by (13), we have that Letting , Since , . Thus, letting , By assumptions, it is clear that .
Since , for all , , and , by (13) and letting , we have Let . Then,
This completes the proof.

Theorem 6. Let and let   be a nonempty subset of . Let be a vector-valued mapping with , for all and let   be a vector-valued mapping. If there exist , , and such that then is a solution of -VEPC, where is a positive real number.

Proof. Let , and suppose that there exist and such that We next will show that is a solution of -VEPC. If not, then there exists such that Since and , we have Thus, by (23), This is a contradiction. Hence, is a solution of -VEPC. This completes the proof.

Next, we consider the following -vector equilibrium problem without constraints (for short -VEP): find such that where is a nonempty subset of , is a vector-valued mapping, , and is a positive real number.

If , , and if is a solution of -VEP, then is a solution of -efficient solution of vector optimization problem of , where is a vector-valued mapping.

Theorem 7. Let and be a nonempty subset of . Let be a vector-valued mapping with , for all . Then is a solution of -VEP if and only if where is a positive real number.

Proof. If is a solution of -VEP, then By Lemma 3(ii), we have
On the other hand, suppose that is not a solution of -VEP. Then, there exists such that By Lemma 3(i), we have This is a contradiction. This completes the proof.

Conflict of Interests

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

Acknowledgments

This research was supported by Project supported by the Key Program of National Natural Science of China (Grant no. 71133007), National Natural Science Foundation of China (Grant no. 71373297), and Program for New Century Excellent Talents in University of Ministry of Education of China (NCET-10-0883).