# Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory

arXiv: https://arxiv.org/abs/2110.02273 · HTML: https://mlsanity.com/paper/2110.02273/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [Bilevel approaches for learning of variational imaging models](https://mlsanity.com/paper/1505.02120/) (2015)
- [Analytical aspects of spatially adapted total variation regularisation](https://mlsanity.com/paper/1609.01074/) (2016)
- [Learning optimal spatially-dependent regularization parameters in total variation image restoration](https://mlsanity.com/paper/1603.09155/) (2016)
- [The structure of optimal parameters for image restoration problems](https://mlsanity.com/paper/1505.01953/) (2015)
- [Bilevel parameter learning for nonlocal image denoising models](https://mlsanity.com/paper/1912.02347/) (2019)
- [Dynamic sampling schemes for optimal noise learning under multiple nonsmooth constraints](https://mlsanity.com/paper/1403.1278/) (2014)
- [Analysis and optimisation of a variational model for mixed Gaussian and Salt & Pepper noise removal](https://mlsanity.com/paper/1809.08991/) (2018)
- [Parameter learning and fractional differential operators: application in image regularization and decomposition](https://mlsanity.com/paper/2001.03394/) (2020)
- [Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization](https://mlsanity.com/paper/2107.08100/) (2021)

## Abstract (arXiv)

We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliar variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and Strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.