Techniques for gradient based bilevel optimization with nonsmooth lower level problems

Abstract: We propose techniques for approximating bilevel optimization problems with non-smooth and non-unique lower level problems. The key is the substitution of the lower level minimization problem with an iterative algorithm that is guaranteed to converge to a minimizer of the problem. Using suitable non-linear proximal distance functions, the update mappings of such an iterative algorithm can be differentiable, notwithstanding the fact that the minimization problem is non-smooth. This technique for smoothly approximating the solution map of the lower level problem raises several questions that are discussed in this paper.