Gradient-Based Solution Algorithms for a Class of Bilevel Optimization and Optimal Control Problems with a Nonsmooth Lower Level

Abstract

The aim of this paper is to explore a peculiar regularization effect that occurs in the sensitivity analysis of certain elliptic variational inequalities of the second kind. The effect causes the solution operator of the variational inequality at hand to be continuously Fréchet differentiable, although the problem itself contains nondifferentiable terms. Our analysis shows in particular that standard gradient-based algorithms can be used to solve bilevel optimization and optimal control problems that are governed by elliptic variational inequalities of the considered type---all without regularizing the nondifferentiable terms in the lower-level problem and without losing desirable properties of the solution such as, e.g., sparsity. Our results can, for instance, be used in the optimal control of Casson fluids and in bilevel optimization approaches for parameter learning in total variation image denoising models.