Journal of Optimization Theory and Applications

, Volume 115, Issue 2, pp 407–417

# Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

• X.Q. Yang
• J.C. Yao
Article

## Abstract

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.

Vector variational inequalities set-valued mappings gap functions existence of a solution semi-infinite programming

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© Plenum Publishing Corporation 2002

## Authors and Affiliations

• X.Q. Yang
• 1
• J.C. Yao
• 2
1. 1.Department of Applied MathematicsHong Kong Polytechnic UniversityKowloonHong Kong
2. 2.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan

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