Signal recovery by minimizing the Moreau envelope over the fixed point set of nonexpansive mappings

Many inverse problems in signal processing have been reduced to a certain nonsmooth convex optimization problem of which the solution set is a closed convex set in a real Hilbert space. This fact indicates that each point in the solution set may differ too largely in some aspects to be ignored. However, almost all existing algorithms designed specially for such a convex optimization problem converge to only an unspecified point in the solution set and cannot pick up a best one, in some additional strategic criterion, among the set. In this paper, we present a flexible utilization of the hybrid steepest descent method for the strategic point selection among the solution set of nonsmooth convex optimization problems. The proposed method can minimize sum of the Moreau envelope of nonsmooth convex functions over the fixed point set of a variety of nonexpansive mappings (e.g., Proximal Forward-Backward splitting operator, Douglas-Rachford splitting operator and Primal-Dual splitting operator) that characterize precisely the solution sets of the nonsmooth convex optimization problems.