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

which is assumed to be nonempty.

Consider the vector equilibrium problems with constraints (for short, VEPC): finding Open image in new window such that

where Open image in new window is a convex cone in Open image in new window .

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

The set of strictly positive functional in Open image in new window is denoted by Open image in new window , that is,

It is well known that

(i)if Open image in new window , then Open image in new window has a base;

(ii)if Open image in new window is a Hausdorff locally convex space, then Open image in new window if and only if Open image in new window has a base;

(iii)if Open image in new window is a separable normed space and Open image in new window is a pointed closed convex cone, then Open image in new window 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 Open image in new window be an arbitrary nonempty subset and Open image in new window . The symbol Open image in new window denotes the closure of Open image in new window , and Open image in new window denotes the generated cone of Open image in new window , that is, Open image in new window . When Open image in new window is a convex, so is Open image in new window .

Remark 2.2.

Obviously, we have

(i) Open image in new window ;

(ii) Open image in new window ;

(iii)if Open image in new window satisfying for all Open image in new window , Open image in new window , then Open image in new window .

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

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

(2)Let Open image in new window be a nonempty subset and let Open image in new window be a convex cone.

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

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

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

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

     
  2. (ii)

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

     
  3. (iii)

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

     

Remark 2.4.

For Open image in new window , the invex set is a convex set and the Open image in new window -preinvex mapping is a convex mapping. However, there are mappings which are Open image in new window -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 Open image in new window . Then Open image in new window is generalized Open image in new window -subconvexlike if and only if Open image in new window is convex.

Lemma 2.6.

Assume that (i) Open image in new window is a nonempty subset and Open image in new window is a convex cone with Open image in new window . (ii) Open image in new window is convex. Then Open image in new window is also convex.

Proof.

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

Note that Open image in new window and the closure of a convex set is convex, then Open image in new window is a convex set. The proof is finished.

Lemma 2.7 (see [1]).

If Open image in new window , Open image in new window , then Open image in new window .

Assume that Open image in new window , a vector Open image in new window is called a weakly efficient solution of (VEPC), if Open image in new window satisfies

Definition 2.8 (see [6]).

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

Remark 2.9.

Obviously, Open image in new window is a globally efficient solution of (VEPC), then Open image in new window 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) Open image in new window and there exists Open image in new window such that Open image in new window ; (ii) Open image in new window is a generalized Open image in new window -subconvexlike on Open image in new window . Then Open image in new window is a globally efficient solution of (VEPC) if and only if there exists Open image in new window and Open image in new window such that

Proof.

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

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

We need to show that Open image in new window .

In fact, if Open image in new window , then Open image in new window . By (3.10), we have

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

Secondly, we show that Open image in new window .

For any Open image in new window , since Open image in new window , then there exists a balanced neighborhood Open image in new window of zero element such that

Note that Open image in new window , and there exists Open image in new window such that Open image in new window

By the arbitrariness of Open image in new window , we have Open image in new window .

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

Taking Open image in new window in (3.10), we get

Thus (3.2) holds.

Then (3.1) holds.

Conversely, if Open image in new window is not a globally efficient solution of (VEPC), then for any pointed convex cone Open image in new window with Open image in new window , we have
Obviously, Open image in new window is a pointed convex cone and Open image in new window . By (3.21), then there exists Open image in new window such that
By the definition of Open image in new window , we get
This together with (3.24) implies that
On the other hand, since Open image in new window , by (3.1) and (3.2), we get

which contradicts (3.26). The proof is finished.

Corollary 3.2.

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

Proof.

Since Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Then Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window . 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 Open image in new window is convex is extended to invex.

(2) Open image in new window is Open image in new window -convex in Open image in new window is extended to Open image in new window -preinvex in Open image in new window

(3) Open image in new window is Open image in new window -convex is extended to Open image in new window -preinvex.

Next, we introduce Gateaux derivative of mapping.

Let Open image in new window and let Open image in new window be a mapping. Open image in new window is called Gateaux differentiable at Open image in new window if for any Open image in new window , there exists limit

Mapping Open image in new window is called Gateaux derivative of Open image in new window at Open image in new window .

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

Theorem 3.4.

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

Proof.

Assume that Open image in new window is a globally efficient solution of (VEPC), by Corollary 3.2, there exists Open image in new window and Open image in new window such that
By (3.32), for any Open image in new window , we have
Since Open image in new window is Gateaux differentiable at Open image in new window , and Open image in new window is Gateaux differentiable at Open image in new window , letting Open image in new window in (3.34), we have
Conversely, if Open image in new window is not a globally efficient solution of (VEPC), a similar proof of (3.24) in Theorem 3.1, there exists Open image in new window such that
This together with Open image in new window being cone yields that
Since Open image in new window is closed, taking Open image in new window in the above formula, we have
Note that Open image in new window , then we have
This together with (3.37) yields that
With similar proof of (3.41), we get
This together with (3.42) implies that

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 Open image in new window be the space of all bounded linear mapping from Open image in new window to Open image in new window . We denote by Open image in new window the value of Open image in new window at Open image in new window .

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

where Open image in new window is a mapping from Open image in new window to Open image in new window .

Definition 4.1 (see [18]).

If Open image in new window , Open image in new window , and if Open image in new window is a globally efficient solution of (VEPC), then Open image in new window is called a globally efficient solution of (VVIC).

Theorem 4.2.

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

Proof.

then Open image in new window is invex with respect to Open image in new window Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex with respect to Open image in new window , and Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Thus the conditions of Theorem 3.4 are satisfied. Note that Open image in new window , 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 Open image in new window is a mapping.

Definition 4.3 (see [18]).

If Open image in new window , Open image in new window , and if Open image in new window is a globally efficient solution of (VEPC), then Open image in new window is called a globally efficient solution of (VOP).

Theorem 4.4.

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

Proof.

then Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex with respect to Open image in new window , and Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Thus the conditions of Theorem 3.4 are satisfied. Note that Open image in new window , 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) Open image in new window and there exists Open image in new window such that Open image in new window ; (ii) Open image in new window is generalized Open image in new window -subconvexlike on Open image in new window . Then Open image in new window is a globally efficient solution of (VOP) if and only if there exists Open image in new window and Open image in new window such that

Notes

Acknowledgment

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

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Copyright information

© Qiusheng Qiu. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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