A nonmonotone extrapolated proximal gradient-subgradient algorithm beyond global Lipschitz gradient continuity

TL;DR

This paper introduces a novel proximal gradient algorithm capable of handling composite optimization problems with non-globally Lipschitz continuous gradients, featuring nonmonotone line search and extrapolation for potential acceleration.

Contribution

It proposes a problem-parameter-free algorithm with nonmonotone line search and extrapolation, extending convergence analysis beyond the global Lipschitz gradient setting without boundedness assumptions.

Findings

The algorithm converges under the Kurdyka-Łojasiewicz property.

Numerical experiments demonstrate the algorithm's effectiveness.

Extrapolation combined with nonmonotone line search improves performance.

Abstract

With the advancement of modern applications, an increasing number of composite optimization problems arise whose smooth component does not possess a globally Lipschitz continuous gradient. This setting prevents the direct use of the proximal gradient (PG) method and its variants, and has motivated a growing body of research on new PG-type methods and their convergence theory, in particular, global convergence analysis without imposing any explicit or implicit boundedness assumptions on the iterates. Until recently, the first complete analysis of this kind has been established for the PG method and its specific nonmonotone variants, which has since stimulated further exploration along this research direction. In this paper, we consider a general composite optimization model beyond the global Lipschitz gradient continuity setting. We propose a novel problem-parameter-free algorithm that incorporates a carefully designed nonmonotone line search to handle the non-global Lipschitz gradient continuity, together with an extrapolation step to achieve potential acceleration. Despite the added technical challenges introduced by combining extrapolation with nonmonotone line search, we establish a refined convergence analysis for the proposed algorithm under the Kurdyka-{\L} ojasiewicz property, without requiring any boundedness assumptions on the iterates. This work thus further advances the theoretical understanding of PG-type methods in the non-global Lipschitz gradient continuity setting. Finally, we conduct numerical experiments to illustrate the effectiveness of our algorithm and highlight the advantages of integrating extrapolation with a nonmonotone line search.