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

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.
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https://hal.archives-ouvertes.fr/hal-00789970
Contributor : Emilie Chouzenoux <>
Submitted on : Tuesday, February 19, 2013 - 10:38:01 AM
Last modification on : Wednesday, February 3, 2021 - 7:54:27 AM
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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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