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arXiv · 2107.08100

Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

Juan Carlos De los Reyes·David Villacís
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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 non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

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Juan Carlos De los Reyes, David Villacís. 2021-07-16. Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization. https://arxiv.org/abs/2107.08100

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