An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions

TL;DR

This paper introduces an inertial forward-backward algorithm for nonconvex optimization, demonstrating convergence to critical points and applying it to image restoration and solution recovery.

Contribution

It presents a novel inertial forward-backward algorithm for nonconvex functions with convergence guarantees under the Kurdyka- inequality.

Findings

Algorithm converges to critical points for semi-algebraic functions.

Effective in recovering local optima in nonconvex problems.

Successful application to noisy image restoration.

Abstract

We propose a forward-backward proximal-type algorithm with inertial/memory effects for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. The sequence of iterates generated by the algorithm converges to a critical point of the objective function provided an appropriate regularization of the objective satisfies the Kurdyka-\L{}ojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions. We illustrate the theoretical results by considering two numerical experiments: the first one concerns the ability of recovering the local optimal solutions of nonconvex optimization problems, while the second one refers to the restoration of a noisy blurred image.