  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.
