# Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

arXiv: https://arxiv.org/abs/2107.08100 · HTML: https://mlsanity.com/paper/2107.08100/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2021.

## Built on by (most cited first, 1 of 1)

- [Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory](https://mlsanity.com/paper/2110.02273/) (2021)

## Abstract (arXiv)

We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.