Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization

TL;DR

This paper introduces a variable smoothing alternating proximal gradient algorithm for nonconvex, nonsmooth optimization problems, demonstrating its efficiency and effectiveness through theoretical analysis and practical experiments.

Contribution

It develops a novel algorithm combining variable smoothing with proximal gradient methods for coupled nonconvex nonsmooth problems, with proven iteration complexity.

Findings

Achieves $ ext{O}( ext{ε}^{-3})$ iteration complexity for stationary points.

Demonstrates superior performance in sparse signal recovery.

Shows effectiveness in image denoising tasks.

Abstract

In this paper, we consider a broad class of nonconvex and nonsmooth optimization problems, where one objective component is a nonsmooth weakly convex function composed with a linear operator. By integrating variable smoothing techniques with first-order methods, we propose a variable smoothing alternating proximal gradient algorithm that features flexible parameter choices for step sizes and smoothing levels. Under mild assumptions, we establish that the iteration complexity to reach an $\varepsilon$-approximate stationary point is $\mathcal{O}(\varepsilon^{-3})$. The proposed algorithm is evaluated on sparse signal recovery and image denoising problems. Numerical experiments demonstrate its effectiveness and superiority over existing algorithms.