Doubly relaxed forward-Douglas--Rachford splitting for the sum of two nonconvex and a DC function

TL;DR

This paper introduces a novel splitting algorithm for structured nonconvex nonsmooth optimization problems involving three functions, including a difference-of-convex component, with proven convergence and competitive performance.

Contribution

The paper proposes a new doubly relaxed forward-Douglas--Rachford splitting algorithm with convergence analysis for nonconvex problems involving DC functions, unifying and extending existing methods.

Findings

Proven subsequential and full convergence to stationary points.

Established convergence rates for iterates and objective values.

Demonstrated competitive performance on matrix completion and estimation tasks.

Abstract

In this paper, we consider a class of structured nonconvex nonsmooth optimization problems whose objective function is the sum of three nonconvex functions, one of which is expressed in a difference-of-convex (DC) form. This problem class covers several important structures in the literature including the sum of three functions and the general DC program. We propose a splitting algorithm and prove the subsequential convergence to a stationary point of the problem. The full sequential convergence, along with convergence rates for both the iterates and objective function values, is then established without requiring differentiability of the concave part. Our analysis not only extends but also unifies and improves recent convergence analyses in nonconvex settings. We benchmark our proposed algorithm with notable algorithms in the literature to show its competitiveness on a low rank matrix completion problem and a simutaneously sparse and low-rank matrix estimation problem. Our algorithm exhibits very competitive results compared to notable algorithms in the literature, on both synthetic data and public dataset.