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 language | English |
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Pages (from-to) | 565-586 |
Number of pages | 22 |
Journal | Journal of Optimization Theory and Applications |
Volume | 83 |
Issue number | 3 |
DOIs | |
Publication status | Published - 1 Dec 1994 |
Externally published | Yes |
Keywords
- cones
- polyhedra
- range sets
- stability
- Variational inequalities
ASJC Scopus subject areas
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics