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Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs

Hugo Raguet (hugo.raguet***at***gmail.com)
Loc Landrieu (landrieu.loic***at***gmail.com)

Abstract: We present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators \sum_{i=1}^n A_i + B with B cocoercive, involving only the computation of B and of the resolvent of each A_i separately. This allows in particular to minimize functionals of the form \sum_{i=1}^n g_i + f with f smooth. By adapting the underlying metric, such preconditioning can serve two practical purposes: first, it might accelerate the convergence, or second, it might simplify the computation of the resolvent of A_i for some i. In addition, in many cases of interest, our preconditioning strategy allows the economy of storage and computation concerning some auxiliary variables. In particular, we show how this approach can handle large-scale, nonsmooth, convex optimization problems structured on graphs, which arises in many image processing or learning applications, and that it compares favorably to alternatives in the literature.

Keywords: preconditioning; forward-backward splitting; monotone operator splitting; proximal splitting; nonsmooth convex optimization; quasi-Newton methods; graph learning; graph sparsity; total variation; Mumford-Shah functional; geoinformatics

Category 1: Convex and Nonsmooth Optimization

Category 2: Convex and Nonsmooth Optimization (Generalized Convexity/Monoticity )


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Entry Submitted: 04/21/2015
Entry Accepted: 04/21/2015
Entry Last Modified: 07/06/2015

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