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

• Published in: Applied mathematics and computation Vol. 392; p. 125719
• Main Authors: Bonettini, S., Prato, M., Rebegoldi, S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2021
• Subjects: Inexact forward–backward methods; Kurdyka–Łojasiewicz property; Nonconvex problems; Numerical optimization; 65K05; Numerical optimization; Kurdyka–Łojasiewicz property; 90C06; 90C90; Inexact forward–backward methods; Nonconvex problems
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125719

•The minimization of nonconvex Kurdyka–Lojasiewicz functions is addressed.•Convergence results on forward-backward methods are provided.•Inexact computation of the proximal gradient point is allowed.•Implementable rules to fulfill the theoretical requirements are discussed.•Numerical tests of image deconvolution problems are shown. 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 with 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–Łojasiewicz 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 Bonettini et al. (2017) 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.