New convergence results for the inexact variable metric forward-backward method

Forward-backward methods are valid tools to solve a variety of optimization problems where the objective function is the sum of a smooth, possibly nonconvex term plus a convex, possibly nonsmooth function. The corresponding iteration is built on two main ingredients: the computation of the gradient of the smooth part and the evaluation of the proximity (or resolvent) operator associated to the convex term. One of the main difficulties, from both implementation and theoretical point of view, arises when the proximity operator is computed in an inexact way. The aim of this paper is to provide new convergence results about forward-backward methods with inexact computation of the proximity operator, under the assumption that the objective function satisfies the Kurdyka-Lojasiewicz property. In particular, we adopt an inexactness criterion which can be implemented in practice, while preserving the main theoretical properties of the proximity operator. The main result is the proof of the convergence of the iterates generated by the forward-backward algorithm in [1] to a stationary point. Convergence rate estimates are also provided. At the best of our knowledge, there exists no other inexact forward-backward algorithm with proved convergence in the nonconvex case and equipped with an explicit procedure to inexactly compute the proximity operator.