Epigraphical splitting for solving constrained convex optimization problems with proximal tools

Abstract : 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 q-norms with q ≥ 1, distance functions to a convex set, and L1,p-norms with p ∈ {2, +∞}. 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. A second application to a pulse shape design problem is provided in order to illustrate the flexibility of the proposed approach.
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Signal, Image and Video Processing, Springer Verlag, 2015, 9 (8), pp.1737-1749. 〈10.1007/s11760-014-0664-1〉
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Giovanni Chierchia, Nelly Pustelnik, Jean-Christophe Pesquet, Béatrice Pesquet-Popescu. Epigraphical splitting for solving constrained convex optimization problems with proximal tools. Signal, Image and Video Processing, Springer Verlag, 2015, 9 (8), pp.1737-1749. 〈10.1007/s11760-014-0664-1〉. 〈hal-01372568〉

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