G\^ateaux semiderivative approach applied to shape optimization for contact problems

TL;DR

This paper develops a Gâteaux semiderivative approach to derive optimality conditions for complex shape optimization problems constrained by contact variational inequalities, addressing non-smoothness and non-convexity.

Contribution

It introduces a novel application of Gâteaux semiderivatives to shape optimization with contact constraints, enabling the formulation of optimality conditions for non-smooth VI-constrained problems.

Findings

Formulated optimality conditions using Gâteaux semiderivatives.

Addressed challenges of non-smooth, non-convex shape optimization.

Applied method to contact problem constraints.

Abstract

Shape optimization problems constrained by variational inequalities (VI) are non-smooth and non-convex optimization problems. The non-smoothness arises due to the variational inequality constraint, which makes it challenging to derive optimality conditions. Besides the non-smoothness there are complementary aspects due to the VIs as well as distributed, non-linear, non-convex and infinite-dimensional aspects due to the shapes which complicate to set up an optimality system and, thus, to develop efficient solution algorithms. In this paper, we consider G\^ateaux semiderivatives in order to formulate optimality conditions. In the application, we concentrate on a shape optimization problem constrained by the contact problem.