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

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.

Mots clés: Nonconvex optimization; Nonsmooth optimization; Majorize-Minimize algorithm; Forward-Backward algorithm; Image reconstruction; Proximity operator

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

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, 2014, 162 (1), pp.107-132. ⟨10.1007/s10957-013-0465-7⟩. ⟨hal-00789970⟩