On the range sets of variational inequalities

Liqun Qi, H. Y. Jiang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

This paper investigates the closedness and convexity of the range sets of the variational inequality (VI) problem defined by an affine mapping M and a nonempty closed convex set K. It is proved that the range set is closed if K is the union of a polyhedron and a compact convex set. Counterexamples are given such that the range set is not closed even if K is a simple geometrical figure such as a circular cone or a circular cylinder in a three-dimensional space. Several sufficient conditions for closedness and convexity of the range set are presented. Characterization for the convex hull of the range set is established in the case where K is a cone, while characterization for the closure of the convex hull of the range set is established in general. Finally, some applications to stability of VI problems are derived.
Original languageEnglish
Pages (from-to)565-586
Number of pages22
JournalJournal of Optimization Theory and Applications
Volume83
Issue number3
DOIs
Publication statusPublished - 1 Dec 1994
Externally publishedYes

Keywords

  • cones
  • polyhedra
  • range sets
  • stability
  • Variational inequalities

ASJC Scopus subject areas

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

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