Nonconvex quasi-variational inequalities: stability analysis and application to numerical optimization

TL;DR

This paper develops a stability analysis framework for nonconvex quasi-variational inequalities using optimal value functions, introduces a semismooth Newton method, and demonstrates its effectiveness on various optimization problems without relying on convexity or higher-order derivatives.

Contribution

It provides the first stability analysis and Newton-type solution method for nonconvex QVIs that do not require convexity or second/third order derivatives.

Findings

New coderivative estimates for nonconvex QVIs.

Robust stability conditions for the solution map.

Effective semismooth Newton method tested on literature examples.

Abstract

We consider a parametric quasi-variational inequality (QVI) without any convexity assumption. Using the concept of \emph{optimal value function}, we transform the problem into that of solving a nonsmooth system of inequalities. Based on this reformulation, new coderivative estimates as well as robust stability conditions for the optimal solution map of this QVI are developed. Also, for an optimization problem with QVI constraint, necessary optimality conditions are constructed and subsequently, a tailored semismooth Newton-type method is designed, implemented, and tested on a wide range of optimization examples from the literature. In addition to the fact that our approach does not require convexity, its coderivative and stability analysis do not involve second order derivatives, and subsequently, the proposed Newton scheme does not need third order derivatives, as it is the case for some previous works in the literature.