Gap functions and existence of solutions to set-valued vector variational inequalities

Xiaoqi Yang, J. C. Yao

Research output: Journal article publicationJournal articleAcademic researchpeer-review

72 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)407-417
Number of pages11
JournalJournal of Optimization Theory and Applications
Issue number2
Publication statusPublished - 1 Nov 2002


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

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

Cite this