On the range sets of variational inequalities

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.