Optimality Conditions of Globally Efficient Solution for Vector Equilibrium Problems with Generalized Convexity

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Research Article

Abstract

We study optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity. The necessary and sufficient conditions of globally efficient solution for the vector equilibrium problems are obtained. The Kuhn-Tucker condition of globally efficient solution for vector equilibrium problems is derived. Meanwhile, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems with constraints.

Keywords

Convex Cone Efficient Solution Topological Vector Space Vector Optimization Problem Vector Variational Inequality
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Throughout the paper, let , and be real Hausdorff topological vector spaces, a nonempty subset, and denotes the zero element of . Let and be two pointed convex cones (see [1]) such that , , where denotes the interior of . Let be a mapping and let be a mapping such that , for all . For each , we denote and define the constraint set

which is assumed to be nonempty.

Consider the vector equilibrium problems with constraints (for short, VEPC): finding such that
(VEPC)

where is a convex cone in .

Vector equilibrium problems, which contain vector optimization problems, vector variational inequality problems, and vector complementarity problems as special case, have been studied by Ansari et al. [2, 3], Bianchi et al. [4], Fu [5], Gong [6], Gong and Yao [7, 8], Hadjisavvas and Schaible [9], Kimura and Yao [10, 11, 12, 13], Oettli [14], and Zeng et al. [15]. But so far, most papers focused mainly on the existence of solutions and the properties of the solutions, there are few papers which deal with the optimality conditions. Giannessi et al. [16] turned the vector variational inequalities with constraints into another vector variational inequalities without constraints. They gave sufficient conditions for efficient solution and weakly efficient solution of the vector variational inequalities in finite dimensional spaces. Morgan and Romaniello [17] gave scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities in Hilbert space by using the concept of subdifferential of the function. Gong [18] presented the necessary and sufficient conditions for weakly efficient solution, Henig efficient solution, and superefficient solution for the vector equilibrium problems with constraints under the condition of cone-convexity. However, the condition of cone-convexity is too strong. Some generalized convexity has been developed, such as cone-preinvexity (see [19]), cone-convexlikeness (see [20]), cone-subconvexlikeness (see [21]), and generalized cone-convexlikeness (see [22]). Among them, the generalized cone-subconvexlikeness has received more attention. Then, it is important to give the optimality conditions for the solution of (VEPC) under conditions of generalized convexity. Moreover, it appears that no work has been done on the Kuhn-Tucker condition of solution for (VEPC). This paper is the effort in this direction.

In the paper, we study the optimality conditions for the vector equilibrium problems. Firstly, we present the necessary and sufficient conditions for globally efficient solution of (VEPC) under generalized cone-subconvexlikeness. Secondly, we prove that the Kuhn-Tucker condition for (VEPC) is both necessary and sufficient under the condition of cone-preinvexity. Meanwhile, we obtain the optimality conditions for vector optimization problems with constraints and vector variational inequality problems with constraints in Section 4.

2. Preliminaries and Definitions

Let , be the dual space of , , respectively, then the dual cone of is defined as
The set of strictly positive functional in is denoted by , that is,

It is well known that

(i)if , then has a base;

(ii)if is a Hausdorff locally convex space, then if and only if has a base;

(iii)if is a separable normed space and is a pointed closed convex cone, then is nonempty (see [1]).

Remark 2.1.

The positive cone in many common Banach spaces possesses strictly positive functionals. However, this is not always the case (see [23]).

Let be an arbitrary nonempty subset and . The symbol denotes the closure of , and denotes the generated cone of , that is, . When is a convex, so is .

Remark 2.2.

Obviously, we have

(iii)if satisfying for all , , then .

Several definitions of generalized convex mapping have been introduced in literature.

(1)Let be a nonempty convex subset and let be a convex cone. A mapping is called -convex, if for all , for all , we have

(2)Let be a nonempty subset and let be a convex cone.

(i)A mapping is called -convexlike (see [20]), if for all , for all , there exists such that
(ii) is said to be -subconvexlike (see [21]), if there exists such that for all , for all , for all , there exists such that
(iii) is said to be generalized -subconvexlike (see [22]), if there exists such that for all , for all , for all , there exists , such that

A nonempty subset is called invex with respect to , if there exists a mapping such that for any , and , .

(3)Let be a invex set with respect to . A mapping is said to be -preinvex with respect to (see [19]), if for any , and , we have
Remark 2.3.
1. (i)

From [21], we know that is -convexlike on if and only if is a convex set and is -subconvexlike on if and only if is a convex set.

2. (ii)

If is a convex set, so is . By Lemma of [24], is convex. This shows that -convexlikeness implies -subconvexlikeness. But in general the converse is not true (see [21]).

3. (iii)

It is clear that -subconvexlikeness implies generalized -subconvexlikeness. But in general the converse is not true (see [22]).

Remark 2.4.

For , the invex set is a convex set and the -preinvex mapping is a convex mapping. However, there are mappings which are -preinvex but not convex (see [25]).

Relationships among various types of convexity are as shown below:

Yang [26] proved the following Lemma in Banach space; Chen and Rong [27] generalized the result to topological vector space.

Lemma 2.5.

Assume that . Then is generalized -subconvexlike if and only if is convex.

Lemma 2.6.

Assume that (i) is a nonempty subset and is a convex cone with . (ii) is convex. Then is also convex.

Proof.

By Lemma 2.5 and Remark 2.1(iii), we deduce that is a convex set. It is not difficult to prove that is a convex set.

Note that and the closure of a convex set is convex, then is a convex set. The proof is finished.

Lemma 2.7 (see [1]).

Assume that , a vector is called a weakly efficient solution of (VEPC), if satisfies

Definition 2.8 (see [6]).

Let be a convex cone. Also, is said to be a globally efficient solution of (VEPC), if there exists a pointed convex cone with such that
(2.10)

Remark 2.9.

Obviously, is a globally efficient solution of (VEPC), then is also a weakly efficient solution of (VEPC). But in general the converse is not true (see [6]).

3. Optimality Conditions

Theorem 3.1.

Assume that (i) and there exists such that ; (ii) is a generalized -subconvexlike on . Then is a globally efficient solution of (VEPC) if and only if there exists and such that

Proof.

Assume that is a globally efficient solution of (VEPC), then there exists a pointed convex cone with such that
Since is a pointed convex cone with , then
Note that , for all and above formula, it is not difficult to prove
Since and are two open sets and are two pointed convex cones, by (3.5), we have
Moreover, since is a generalized -subconvexlike on , by Lemma 2.5, is convex. This follows from Lemma 2.6 that is convex. By the standard separation theorem (see [1, page 76]), there exists such that
Since is a cone, it follows from (3.7) that
Note that , thus . By (3.8), we obtain immediately
It implies that
(3.10)
On the other hand, by and (3.7), we get
(3.11)
Since for all , for all , we have , by (3.11), we get
(3.12)
Letting , we have
(3.13)
Firstly, we prove that
(3.14)

Since is convex and is nonempty, then . Note that and (3.13), and we have . With similar proof of , we can prove that .

We need to show that .

In fact, if , then . By (3.10), we have
(3.15)

On the other hand, since , , by Lemma 2.7, we have , which is a contradiction with (3.15).

Secondly, we show that .

For any , since , then there exists a balanced neighborhood of zero element such that
(3.16)

Note that , and there exists such that

Since , then
(3.17)

By the arbitrariness of , we have .

Lastly, we show that (3.1) and (3.2) hold.

Taking in (3.10), we get
(3.18)
Moreover, since , then
(3.19)

Thus (3.2) holds.

Since and , by (3.10), we have
(3.20)

Then (3.1) holds.

Conversely, if is not a globally efficient solution of (VEPC), then for any pointed convex cone with , we have
(3.21)
By , let
(3.22)
Obviously, is a pointed convex cone and . By (3.21), then there exists such that
(3.23)
By the definition of , we get
(3.24)
Moreover, since and , then
(3.25)
This together with (3.24) implies that
(3.26)
On the other hand, since , by (3.1) and (3.2), we get
(3.27)

which contradicts (3.26). The proof is finished.

Corollary 3.2.

Assume that (i) is invex with respect to ; (ii) and there exists such that ; (iii) is -preinvex on with respect to , and is -preinvex on with respect to . Then is a globally efficient solution of (VEPC) if and only if there exist and such that (3.1) and (3.2) hold.

Proof.

Since is -preinvex on with respect to is -preinvex on with respect to . Then is -preinvex on with respect to . Thus by Theorem 3.1, the conclusion of Corollary 3.2 holds.

Remark 3.3.

Corollary 3.2 generalizes and improves the recent results of Gong (see [18, Theorem  3.3]). Especially, Corollary 3.2 generalizes and improves in the following several aspects.

(1)The condition that the subset is convex is extended to invex.

(2) is -convex in is extended to -preinvex in

(3) is -convex is extended to -preinvex.

Next, we introduce Gateaux derivative of mapping.

Let and let be a mapping. is called Gateaux differentiable at if for any , there exists limit
(3.28)

Mapping is called Gateaux derivative of at .

The following theorem shows that the Kuhn-Tucker condition for (VEPC) is both necessary and sufficient.

Theorem 3.4.

Assume that (i) , are closed, is invex with respect to ; (ii) and there exists such that ; (iii) is -preinvex on with respect to and Gateaux differentiable at , and is Gateaux differentiable at and -preinvex on with respect to ; . Then is a globally efficient solution of (VEPC) if and only if there exists and such that
(3.29)
(3.30)

Proof.

Assume that is a globally efficient solution of (VEPC), by Corollary 3.2, there exists and such that
(3.31)
(3.32)
Since is invex with respect to , then for any ,
(3.33)
By (3.32), for any , we have
(3.34)
Since is Gateaux differentiable at , and is Gateaux differentiable at , letting in (3.34), we have
(3.35)
Conversely, if is not a globally efficient solution of (VEPC), a similar proof of (3.24) in Theorem 3.1, there exists such that
(3.36)
Since , thus we have
(3.37)
Moreover, since is -preinvex on with respect to , then for any , , we have
(3.38)
This together with being cone yields that
(3.39)
Since is closed, taking in the above formula, we have
(3.40)
Note that , then we have
(3.41)
This together with (3.37) yields that
(3.42)
Moreover, since , and , then we have
(3.43)
With similar proof of (3.41), we get
(3.44)
This together with (3.42) implies that
(3.45)

which contradicts (3.29). The proof is finished.

4. Application

As interesting applications of the results of Section 3, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems.

Let be the space of all bounded linear mapping from to . We denote by the value of at .

Equation (VEPC) includes as a special case a vector variational inequality with constraints (for short, (VVIC)) involving

where is a mapping from to .

Definition 4.1 (see [18]).

If , , and if is a globally efficient solution of (VEPC), then is called a globally efficient solution of (VVIC).

Theorem 4.2.

Assume that (i) , are closed, is a nonempty convex subset; (ii) and there exists such that ; (iii) is Gateaux differentiable at and -convex on . Then is a globally efficient solution of (VVIC) if and only if there exists and such that

Proof.

Let

then is invex with respect to is Gateaux differentiable at and -preinvex with respect to , and is Gateaux differentiable at and -preinvex on with respect to . Thus the conditions of Theorem 3.4 are satisfied. Note that , by Theorem 3.4, then the conclusion of Theorem 4.2 holds.

Another special case of (VEPC) is the vector optimization problem with constraints (for short, VOP):
involving

where is a mapping.

Definition 4.3 (see [18]).

If , , and if is a globally efficient solution of (VEPC), then is called a globally efficient solution of (VOP).

Theorem 4.4.

Assume that (i) , are closed, is invex with respect to ; (ii) and there exists such that ; (iii) is Gateaux differentiable at and -preinvex on with respect to , and is Gateaux differentiable at and -preinvex on with respect to . Then is a globally efficient solution of (VOP) if and only if there exists and such that

Proof.

Let

then is Gateaux differentiable at and -preinvex with respect to , and is Gateaux differentiable at and -preinvex on with respect to . Thus the conditions of Theorem 3.4 are satisfied. Note that , by Theorem 3.4, then the conclusion of Theorem 4.4 holds.

By Theorem 3.1, we have the following result.

Theorem 4.5.

Assume that (i) and there exists such that ; (ii) is generalized -subconvexlike on . Then is a globally efficient solution of (VOP) if and only if there exists and such that

Notes

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 10771228).

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