Variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function

Emilie Chouzenoux 1 Jean-Christophe Pesquet 1 Audrey Repetti 1
1 LIGM - Laboratoire d'Informatique Gaspard-Monge
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
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Article dans une revue
Journal of Optimization Theory and Applications, Springer Verlag, 2014, 162 (1), pp.107-132. 〈10.1007/s10957-013-0465-7〉
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Emilie Chouzenoux, Jean-Christophe Pesquet, Audrey Repetti. Variable metric forward-backward algorithm for minimizing the sum of a differentiable function and a convex function. Journal of Optimization Theory and Applications, Springer Verlag, 2014, 162 (1), pp.107-132. 〈10.1007/s10957-013-0465-7〉. 〈hal-00789970〉

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