The Compression Property for Affine Variational Inequalities

Abstract

In 1965, Gale and Nikaidô showed that for any n × n P-matrix A, the only nonnegative vector that A sends into a nonpositive vector is the origin. They applied that result to derive various results including univalence properties of certain nonlinear functions. In this article, we show that an extension of their result holds with the nonnegative orthant replaced by any nonempty polyhedral convex cone. In place of the P-matrix condition, we require a determinantal condition that we call the compression property. When the polyhedral convex cone is the nonnegative orthant, the compression property reduces to the property of being a P-matrix and we recover the Gale-Nikaidô result. We apply the extended theorem to derive tools useful in the analysis of affine variational inequalities over polyhedral convex cones.

Keywords:

• Dual variational inequality
• Monotone graph
• Nonsmooth equations
• Normal manifold
• Normal map
• Variational condition
• Variational inequality

Mathematics Subject Classification:

• Primary: 90C33
• Secondary: 90C31

Acknowledgments

Part of the special issue, “Variational Analysis and Applications.”