Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators

TL;DR

This paper develops a new elementary approach to analyze the directional differentiability of solution maps for variational inequalities with locally Lipschitz continuous operators, including non-unique solutions and weak topologies.

Contribution

It provides a necessary and sufficient criterion for directional differentiability of solution maps in Banach and Hilbert spaces, extending classical results without relying on Mosco convergence.

Findings

Established a criterion for directional differentiability of solution maps.

Applied the results to elastoplasticity, prox-regular sets, and bang-bang control.

Extended analysis to non-unique solutions and weak topology derivatives.

Abstract

This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). In contrast to classical results, our method of proof does not rely on Attouch's theorem on the characterization of Mosco convergence but is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-$\star$ topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang-bang optimal control problem.