Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-{\L}ojasiewicz Property without Global Lipschitz Assumptions

TL;DR

This paper proves that local Lipschitz continuity combined with the Kurdyka-{ extL}ojasiewicz property is sufficient for convergence of the proximal gradient method, removing the need for a global Lipschitz assumption.

Contribution

It demonstrates that global Lipschitz continuity is not necessary for convergence, broadening the applicability of the proximal gradient method.

Findings

Convergence results hold under local Lipschitz and Kurdyka-{ extL}ojasiewicz conditions.

Global Lipschitz assumption can be replaced with local Lipschitz in convergence analysis.

Theoretical framework extends the applicability of proximal gradient methods.

Abstract

We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem numerically. The corresponding global convergence and local rate-of-convergence theory typically assumes, besides some technical conditions, that the smooth function has a globally Lipschitz continuous gradient and that the objective function satisfies the Kurdyka-{\L}ojasiewicz property. Though this global Lipschitz assumption is satisfied in several applications where the objective function is, e.g., quadratic, this requirement is very restrictive in the non-quadratic case. Some recent contributions therefore try to overcome this global Lipschitz condition by replacing it with a local one, but, to the best of our knowledge, they still require some extra condition in order to obtain the desired global and rate-of-convergence results. The aim of this paper is to show that the local Lipschitz assumption together with the Kurdyka-{\L}ojasiewicz property is sufficient to recover these convergence results.