Variable metric inexact line-search based methods for nonsmooth optimization

TL;DR

This paper introduces a variable metric inexact proximal-gradient method for nonsmooth, possibly nonconvex optimization, providing convergence guarantees and demonstrating competitive numerical performance on image restoration tasks.

Contribution

It proposes a novel variable metric inexact proximal-gradient algorithm with convergence analysis for nonconvex and convex problems, including rate estimates.

Findings

Convergence to stationary points in nonconvex case.

Whole sequence convergence to minimizer in convex case.

O(1/k) convergence rate for function values.

Abstract

We develop a new proximal-gradient method for minimizing the sum of a differentiable, possibly nonconvex, function plus a convex, possibly non differentiable, function. The key features of the proposed method are the definition of a suitable descent direction, based on the proximal operator associated to the convex part of the objective function, and an Armijo-like rule to determine the step size along this direction ensuring the sufficient decrease of the objective function. In this frame, we especially address the possibility of adopting a metric which may change at each iteration and an inexact computation of the proximal point defining the descent direction. For the more general nonconvex case, we prove that all limit points of the iterates sequence are stationary, while for convex objective functions we prove the convergence of the whole sequence to a minimizer, under the assumption that a minimizer exists. In the latter case, assuming also that the gradient of the smooth part of the objective function is Lipschitz, we also give a convergence rate estimate, showing the O(1/k) complexity with respect to the function values. We also discuss verifiable sufficient conditions for the inexact proximal point and we present the results of a numerical experience on a convex total variation based image restoration problem, showing that the proposed approach is competitive with another state-of-the-art method.