Optimality conditions for bilevel imaging learning problems with total variation regularization

De los Reyes, Juan Carlos, Villacís, David

SIAM Journal on Imaging Sciences, volume 15, number 4, pages 1646-1689, December 2022, doi: 10.1137/21M143412X

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Abstract

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 nonsmooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

Bibtex

@article{doi:10.1137/21M143412X,
  author   = {De los Reyes, Juan Carlos and Villacís, David},
  title    = {Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization},
  journal  = {SIAM Journal on Imaging Sciences},
  volume   = {15},
  number   = {4},
  pages    = {1646-1689},
  year     = {2022},
  doi      = {10.1137/21M143412X},
  url      = {https://doi.org/10.1137/21M143412X},
  eprint   = {https://doi.org/10.1137/21M143412X},
  abstract = { 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 nonsmooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings. }
}