From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

TL;DR

This paper studies the sensitivity and differentiability of solutions to generalized equations involving monotone operators, providing conditions for directional differentiability and applying results to quasi-generalized equations.

Contribution

It establishes the directional differentiability of the solution map for generalized equations with monotone operators, linking it to the differentiability of constituent mappings.

Findings

Directional differentiability of the solution map verified via single-valued and resolvent differentiability.

Application to quasi-generalized equations with additional solution dependence.

Framework applicable to a broad class of monotone operator problems.

Abstract

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.