Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control

TL;DR

This paper establishes Newton differentiability of solution operators for obstacle-type variational inequalities, enabling efficient semismooth Newton methods for related optimal control problems with proven superlinear convergence.

Contribution

It introduces a novel analysis showing Newton differentiability of solution operators in obstacle variational inequalities, facilitating advanced numerical solution techniques.

Findings

Superlinear convergence of the semismooth Newton method is proven.

Mesh independence of the algorithm is demonstrated through numerical experiments.

The approach is applicable to infinite-dimensional optimal control problems.

Abstract

We prove that solution operators of elliptic obstacle-type variational inequalities (or, more generally, locally Lipschitz continuous functions possessing certain pointwise-a.e. convexity properties) are Newton differentiable when considered as maps between suitable Lebesgue spaces and equipped with the strong-weak Bouligand differential as a generalized set-valued derivative. It is shown that this Newton differentiability allows to solve optimal control problems with H1-cost terms and one-sided pointwise control constraints by means of a semismooth Newton method. The superlinear convergence of the resulting algorithm is proved in the infinite-dimensional setting and its mesh independence is demonstrated in numerical experiments. We expect that the findings of this paper are also helpful for the design of numerical solution procedures for quasi-variational inequalities and the optimal control of obstacle-type variational problems.