Epigraphical projection and proximal tools for solving constrained convex optimization problems

We propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower-level set of a sum of convex functions evaluated over different blocks of the linearly transformed signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower-level set into as many epigraphs as functions involved in the sum. In particular, we focus on constraints involving -norms with , distance functions to a convex set, and -norms with . The proposed approach is validated in the context of image restoration by making use of constraints based on Non-Local Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems.