# Efficient Techniques for Shape Optimization with Variational Inequalities using Adjoints

## TL;DR

This paper develops efficient shape optimization techniques for obstacle problems constrained by variational inequalities, establishing adjoint existence, deriving shape derivatives, and demonstrating convergence with numerical validation.

## Contribution

It introduces a novel approach to handle shape optimization with variational inequalities using adjoints, including existence proofs, derivative formulas, and an effective algorithm.

## Key findings

- Existence of adjoints for regularized problems
- Convergence of solutions to unregularized limits
- Numerical validation of the optimization algorithm

## Abstract

In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for semi-smooth shape optimization problems. In this paper, we consider shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems. Under appropriate assumptions, we prove existence of adjoints for regularized problems and convergence to limiting objects of the unregularized problem. Moreover, we derive existence and closed form of shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically.