Techniques for Gradient Based Bilevel Optimization with Nonsmooth Lower   Level Problems
  We propose techniques for approximating bilevel optimization problems with non-smooth lower level problems that can have a non-unique solution. To this end, we substitute the expression of a minimizer 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. 
Peter Ochs, Ren\'e Ranftl, Thomas Brox, Thomas Pock
math.OC