- Research Article
- Open Access

# Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems

- Zhen-Fei Wei
^{1}and - Xun-Hua Gong
^{1}Email author

**2010**:842715

https://doi.org/10.1155/2010/842715

© Z.-F.Wei and X.-H. Gong. 2010

**Received:**20 January 2010**Accepted:**17 April 2010**Published:**23 May 2010

## Abstract

By using the concept of Fréchet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.

## Keywords

- Convex Cone
- Efficient Solution
- Vector Optimization Problem
- Vector Variational Inequality
- Vector Equilibrium Problem

## 1. Introduction

Recently, some authors have studied the optimality conditions for vector variational inequalities. Giannessi et al. [1] turned the vector variational inequalities with constraints into vector variational inequalities without constraints and gave sufficient conditions for the efficient solutions and the weakly efficient solutions to the vector variational inequalities in . By using the concept of subdifferential of the function, Morgan and Romaniello [2] gave the scalarization and Kuhn-Tucker-like conditions for the weak vector generalized quasivariational inequalities in Hilbert space. Yang and Zheng [3] gave the optimality conditions for approximate solutions of vector variational inequalities in Banach space. On the other hand, some authors have derived the optimality conditions for weakly efficient solutions to vector optimization problems (see [4–19]).

Vector variational inequality problems and vector optimization problems, as well as several other problems, are special realizations of vector equilibrium problems (see [20, 21]); therefore, it is important to give the optimality conditions for the solution to the vector equilibrium problems for in this way we can turn the vector equilibrium problem with constraints to a corresponding scalar optimization problem without constraints, and we can then determine if the solution of the scalar optimization problem is a solution of the original vector equilibrium problem. Under the assumption of convexity, Gong [22] investigated optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to vector equilibrium problems with constraints and obtained that the weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems with constraints are equivalent to solution of corresponding scalar optimization problems without constraints, respectively. Qiu [23] presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong [24] weakened the convexity assumptions in [22] and obtained necessary and sufficient conditions for weakly efficient solution, too.

In this paper, by using the concept of Fréchet differentiability of mapping, we study the optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to the vector equilibrium problems. We give Kuhn-Tucker necessary conditions to the vector equilibrium problems without convexity conditions and Kuhn-Tucker sufficient conditions with convexity conditions.

## 2. Preliminaries and Definitions

Throughout this paper, let be a real normed space, let and be real partially ordered normed spaces, and let and be closed convex pointed cones with , where denotes the interior of the set .

Denote the closure of by and interior of by .

A nonempty convex subset of the convex cone is called a base of , if and . It is easy to see that if and only if has a base.

By the separation theorem of convex sets, we know .

Denote the closed unit ball of by . Suppose that has a base . Let . It is clear that . The will be used for the rest of the paper. For any , denote cone , then cl is a closed convex pointed cone, and , for all (see [25]).

Let be a nonempty open convex subset, and let , be mappings.

where is a convex cone in .

Definition 2.1.

is called a weakly efficient solution to the VEPC.

Definition 2.2 (see [22]).

Definition 2.3 (see [22]).

Definition 2.4 (see [22]).

where is closed unit ball of .

Definition 2.5.

exists, then is called the Gâteaux derivative of at in the direction . If this limit exists for each direction , the mapping is called Gâteaux differentiable at .

Definition 2.6.

then is called the Fréchet derivative of at and is called Fréchet differentiable at .

Remark 2.7.

By [26, Lemma .18], we can see that if is Fréchet differentiable at , then is Gâteaux differentiable at and the Fréchet derivative of at is equal to the Gâteaux derivative of at in each direction .

Definition 2.8.

Lemma 2.9 (see [18]).

Assume that pointed convex cone has a base ; then one has the following.

(i)For any , .

(ii)For any , there exists some with .

(iii) , and when is bounded and closed, then , where is the interior of in with respect to the norm of .

## 3. Optimality Condition

In this section, we give the Kuhn-Tucker necessary conditions and Kuhn-Tucker sufficient conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to the vector equilibrium problems with constraints.

By the proof of Theorem .20 in [26] or from the definition, we have the following lemma.

Lemma 3.1.

Theorem 3.2.

Proof.

for all , ,

This completes the proof.

Remark 3.3.

The condition that there exists such that is a generalized Slater constraint condition. In fact, from the proof of Theorem 3.2, we obtain that there exists such that and .

Theorem 3.4.

then is a weakly efficient solution to the VEPC.

Proof.

This contradicts (3.30). Hence, is a weakly efficient solution to the VEPC.

From Theorems 3.2 and 3.4, and Remark 2.7, we get the following corollary.

Corollary 3.5.

The concept of weakly efficient solution to the vector equilibrium problem requires the condition that the ordering cone has an nonempty interior. If the ordering cone has an empty interior, we cannot discuss the property of weakly efficient solutions to the vector equilibrium problem. However, if the ordering cone has a base, we can give necessary conditions and sufficient conditions for Henig efficient solutions and globally efficient solutions to the vector equilibrium problems with constraints.

Theorem 3.6.

Proof.

This completes the proof.

Theorem 3.7.

then is a Henig efficient solution to the VEPC.

Proof.

This contradicts (3.45). Hence, is a Henig efficient solution to the VEPC.

Corollary 3.8.

Remark 3.9.

If has a bounded closed base , in view of Lemma 2.9, we have ; besides, is a superefficient solution to the VEPC if and only if is a Henig efficient solution to the VEPC (see [22]). Hence, by Corollary 3.8, we have the following corollary.

Corollary 3.10.

Theorem 3.11.

Proof.

for all , ,

This completes the proof.

Theorem 3.12.

then is a globally efficient solution to the VEPC.

Proof.

This contradicts (3.65). Hence, is a globally efficient solution to the VEPC.

Corollary 3.13.

## Declarations

### Acknowledgments

This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province (2008GZS0072), China.

## Authors’ Affiliations

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