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A Block Coordinate Variable Metric Forward-Backward Algorithm

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

Abstract: A number of recent works have emphasized the prominent role played by the Kurdyka-Lojasiewicz inequality for proving the convergence of iterative algorithms solving possibly nonsmooth/nonconvex optimization problems. In this work, we consider the minimization of an objective function satisfying this property, which is a sum of a non necessarily convex differentiable function and a non necessarily differentiable or convex function. The latter function is expressed as a separable sum of functions of blocks of variables. Such an optimization problem can be addressed with the Forward-Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize-Minimize principle. We propose to combine the latter acceleration technique with an alternating minimization strategy which relies upon a flexible update rule. We give conditions under which the sequence generated by the resulting Block Coordinate Variable Metric Forward-Backward algorithm converges to a critical point of the objective function. An application example to a nonconvex phase retrieval problem encountered in signal/image processing shows the efficiency of the proposed optimization method.

Keywords: Nonconvex optimization ; Nonsmooth optimization ; Proximity operator ; Majorize-Minimize algorithm ; Block coordinate descent ; Alternating minimization ; Phase retrieval ; Inverse problems

Category 1: Convex and Nonsmooth Optimization

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


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Entry Submitted: 12/20/2013
Entry Accepted: 12/20/2013
Entry Last Modified: 04/05/2014

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