A four-operator splitting algorithm for nonconvex and nonsmooth optimization

TL;DR

This paper introduces a novel four-operator splitting algorithm for complex nonconvex nonsmooth optimization problems, extending existing methods and demonstrating convergence and practical effectiveness.

Contribution

It extends the Davis-Yin splitting algorithm to handle four-term nonconvex nonsmooth problems with improved convergence guarantees and larger stepsizes.

Findings

Global subsequential convergence to stationary points

Convergence rate of $1/T$ for stationarity measure

Experimental validation of algorithm's effectiveness

Abstract

In this work, we address a class of nonconvex nonsmooth optimization problems where the objective function is the sum of two smooth functions (one of which is proximable) and two nonsmooth functions (one proper, closed and proximable, and the other continuous and weakly concave). We introduce a new splitting algorithm that extends the Davis-Yin splitting (DYS) algorithm to handle such four-term nonconvex nonsmooth problems. We prove that with appropriately chosen stepsizes, our algorithm exhibits global subsequential convergence to stationary points with a stationarity measure converging at a global rate of $1/T$, where $T$ is the number of iterations. When specialized to the setting of the DYS algorithm, our results allow for larger stepsizes compared to existing bounds in the literature. Experimental results demonstrate the practical applicability and effectiveness of our proposed algorithm.