Variable Metric Forward-Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

We consider the minimization of a function G defined on R , 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 KurdykaLojasiewicz 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.