Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems
© Z.-F.Wei and X.-H. Gong. 2010
Received: 20 January 2010
Accepted: 17 April 2010
Published: 23 May 2010
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.
Recently, some authors have studied the optimality conditions for vector variational inequalities. Giannessi et al.  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  gave the scalarization and Kuhn-Tucker-like conditions for the weak vector generalized quasivariational inequalities in Hilbert space. Yang and Zheng  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  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  presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong  weakened the convexity assumptions in  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
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 ).
is called a weakly efficient solution to the VEPC.
Definition 2.2 (see ).
Definition 2.3 (see ).
Definition 2.4 (see ).
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 .
Lemma 2.9 (see ).
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  or from the definition, we have the following lemma.
This completes the proof.
From Theorems 3.2 and 3.4, and Remark 2.7, we get the following corollary.
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.
This completes the proof.
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 ). Hence, by Corollary 3.8, we have the following corollary.
This completes the proof.
This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province (2008GZS0072), China.
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