Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

TLDR

Variational inequality problems over polyhedral convex sets, both linear and nonlinear, are analyzed parametrically. The study aims to characterize Robinson’s strong regularity for such variational inequalities using a new critical‑face condition and to provide a new characterization for the KKT‑associated variational inequality. The authors characterize strong regularity via a critical‑face condition, analyze its implications for complementarity problems, and apply the results to nonlinear programming with canonical perturbations. They obtain a new characterization of strong regularity for the variational inequality arising from KKT conditions.

Abstract

Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson's notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critical face” condition and in other ways. The consequences for complementarity problems are worked out as a special case. Application is also made to standard nonlinear programming problems with parameters that include the canonical perturbations. In that framework a new characterization of strong regularity is obtained for the variational inequality associated with the Karush–Kuhn–Tucker conditions.