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Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

Abstract:

We address the problem of optimal scale-dependent parameter learning in total variation image de-noising. 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 ver-ifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution op-erator. 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.