Techniques for gradient based bilevel optimization with nonsmooth lower level problems

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.

@article{ORBP16,
  title        = {Techniques for gradient based bilevel optimization with nonsmooth lower level problems},
  author       = {P. Ochs and R. Ranftl and T. Brox and T. Pock},
  year         = {2016},
  journal      = {Journal of Mathematical Imaging and Vision},
  number       = {2},
  volume       = {56},
  pages        = {175--194}
}