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We study a bilevel semi-infinite optimization approach for spatially dependent parameter learning in total generalized variation image denoising. We present some analytical results like the existence of solutions for the bilivel problem and the Fréchet differentiability of the solution operator, which allows to prove existence of Lagrange multipliers. In addition, we prove existence of the adjoint state which allows to obtain a gradient characterization. The multipliers associated with the constraints are regular Borel measures which are very difficult to compute. In order to overcome this issue, we propose Moreau-Yosida regularization, where the optimality system associated with the regularized problem is derived and we prove that the solutions of the regularized problems converge to the solution of the original one. The proposed numerical strategy is to use a second-order quasi-Newton method, specifically the BFGS method, together with the Newton Semi-Smooth algorithms for the resolution of TGV image denoising model.
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) .