Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

Zhi-Bin Liu,Jongkyu Kim,Nan-Jing Huang

doi:10.1155/2008/678014

Abstract

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

Highlights

The concept of vector variational inequality was first introduced by Giannessi 1 in 1980
Vector variational inequalities and their generalizations have been used as a tool to solve vector optimization problems see 7, 10–14
Chen and Craven 11 obtained a sufficient condition for the existence of weakly efficient solutions for differentiable vector optimization problems involving differentiable convex functions by using vector variational inequalities for vector valued functions

Summary

Introduction

The concept of vector variational inequality was first introduced by Giannessi 1 in 1980. Kazmi 12 proved a sufficient condition for the existence of weakly efficient solutions for vector optimization problems involving differentiable preinvex functions by using vector variational-like inequalities. Lee et al 7 established the existence of the weakly efficient solution for nondifferentiable vector optimization problems by using vector variational-like inequalities for set-valued mappings. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem without the restrict condition Rm ⊂ C x for all x ∈ Rn. Our results generalize and improve the results obtained by Lee et al 7 and Ansari and Yao 10

Preliminaries

Main results