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Variable Metric Forward-Backward algorithm for minimizing the sum of a differentiable function and a convex function

Emilie Chouzenoux (emilie.chouzenoux***at***univ-mlv.fr)
Jean-Christophe Pesquet (pesquet***at***univ-mlv.fr)
Audrey Repetti (audrey.repetti***at***univ-mlv.fr)

Abstract: We consider the minimization of a function $G$ defined on $R^N$, which is the sum of a (non necessarily convex) differentiable function and a (non necessarily differentiable) convex function. Moreover, we assume that $G$ satisfies the Kurdyka-Lojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize-Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward-Backward algorithm converges to a critical point of $G$. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.

Keywords: Nonconvex optimization ; Nonsmooth optimization ; Majorize-Minimize algorithm ; Forward-Backward algorithm ; Image reconstruction ; Proximity operator

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 01/29/2013
Entry Accepted: 01/31/2013
Entry Last Modified: 10/24/2013

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