Constraint Nondegeneracy in Variational Analysis

Abstract

This paper studies the sensitivity analysis of variational conditions defined over perturbed systems of finitely many nonlinear inequalities or equations, subject to additional fixed polyhedral constraints. If the system of constraints obeys a certain property called nondegeneracy, we show how to construct a local diffeomorphism of the feasible set to its tangent cone. Moreover, this diffeomorphism varies smoothly as the perturbation parameter changes. The original variational condition is then locally equivalent to a variational inequality defined over this (polyhedral convex) tangent cone. This extends stability results already known for variational inequalities over polyhedral convex sets to a substantially more general case. We also show that existence, local uniqueness, and Lipschitz continuity, as well as B-differentiability of the solution, can all be predicted from a single affine variational inequality that is easily computable in terms of the data of the unperturbed problem at the point in question.