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1 LIGM - Laboratoire d-Informatique Gaspard-Monge

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

Author: Emilie Chouzenoux - Jean-Christophe Pesquet - Audrey Repetti -

Source: https://hal.archives-ouvertes.fr/


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