A non-smooth trust-region method for locally Lipschitz functions with application to optimization problems constrained by variational inequalities

TL;DR

This paper introduces a nonsmooth trust-region method tailored for locally Lipschitz functions, specifically addressing variational inequality constraints, with proven convergence and demonstrated effectiveness through numerical experiments.

Contribution

It develops a novel nonsmooth trust-region algorithm with convergence guarantees for variational inequality constrained problems, including a computable model and subdifferential characterization.

Findings

Convergence to C-stationary points is established.

The method effectively handles variational inequality constraints.

Numerical experiments validate the method's performance.

Abstract

We propose a nonsmooth trust-region method for solving optimization problems with locally Lipschitz continuous functions, with application to problems constrained by variational inequalities of the second kind. Under suitable assumptions on the model functions, convergence of the general algorithm to a C-stationary point is verified. For variational inequality constrained problems, we are able to properly characterize the Bouligand subdifferential of the reduced cost function and, based on that, we propose a computable trust-region model which fulfills the convergence hypotheses of the general algorithm. The article concludes with the experimental study of the main properties of the proposed method based on two different numerical instances.