Bilevel Imaging Learning with Total Variation Regularization: Optimality Conditions and Trust-Region Solution Algorithms.

David Villacís

December 2022

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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 can 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 experimental settings.

Bibtex

@phdthesis{thesis,
  author    = {David Villacís},
  title     = {Bilevel Imaging Learning with Total Variation Regularization: Optimality Conditions and Trust-Region Solution Algorithms.},
  school    = {Escuela Politécnica Nacional},
  year      = {2022},
  url       = {https://bibdigital.epn.edu.ec/handle/15000/23455},
  timestamp = {Thu, 01 Apr 2021 15:24:56 +0200},
  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 can 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 experimental settings.}
}