A block coordinate variable metric forward-backward algorithm

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.
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Submitted on : Thursday, February 13, 2014 - 11:25:33 AM
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Emilie Chouzenoux, Jean-Christophe Pesquet, Audrey Repetti. A block coordinate variable metric forward-backward algorithm. Journal of Global Optimization, Springer Verlag, 2016, pp.1-29. ⟨10.1007/s10898-016-0405-9⟩. ⟨hal-00945918⟩

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