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We study bilevel optimization problems restricted to variational inequalities of the second kind with application to image denoising tasks. Several learning problems in image processing can be formulated in the context of a bilevel optimization problem. In such problems the lower level corresponds to a variational formulation for a specific imaging task (denoising, inpainting, segmentation, etc.) and the higher level problem uses the solution provided by the lower level to compute a quality measure using a training dataset. Even though previous work has been proposed by using a smoothed version of the lower level problem in order to characterize an optimality condition, we will focus our efforts on a non-smoothed version of the non-differentiable terms. We will find optimality conditions of the bilevel problem by exploting the directional differentiability of the solution operator of the lower level problem and propose a numeric scheme based on a Trust Region algorithm in order to find candidate solutions.
This presentation is part of Minisymposium “MS62 - Imaging models with non-linear constraints (2 parts)”
organized by: Tuomo Valkonen (University of Liverpool) , Juan Carlos De Los Reyes (Escuela Politécnica Nacional) .