Numerical Optimization Methods for the Optimal Control of Elliptic Variational Inequalities

Abstract

The optimal control of variational inequalities introduces a number of additional challenges to PDE-constrained optimization problems both in terms of theory and algorithms. The purpose of this article is to first introduce the theoretical underpinnings and then to illustrate various types of numerical methods for the optimal control of variational inequalities. For a generic problem class, sufficient conditions for the existence of a solution are discussed and subsequently, the various types of multiplier-based optimality conditions are introduced. Finally, a number of function-space-based algorithms for the numerical solution of these control problems are presented. This includes adaptive methods based on penalization or regularization as well as non-smooth approaches based on tools from non-smooth optimization and set-valued analysis. A new type of projected subgradient method based on an approximation of limiting coderivatives is proposed. Moreover, several existing methods are extended to include control constraints. The computational performance of the algorithms is compared and contrasted numerically.