Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization
Xian-Jun Long, Kang Zeng, Gao-Xi Li, Minh N. Dao, Zai-Yun Peng
TL;DR
This work addresses nonconvex nonsmooth composite optimization with a weakly convex nonsmooth term composed with a linear operator. It introduces a variable smoothing alternating proximal gradient method that leverages a Moreau envelope $g_{\mu}$ to form a smooth surrogate and perform efficient first-order updates on $x$ and $y$. Under standard smoothness and weak convexity assumptions, it proves an iteration complexity of $\mathcal{O}(\varepsilon^{-3})$ to achieve an $\varepsilon$-approximate stationary point. Numerical experiments on sparse signal recovery and image denoising demonstrate that the proposed method outperforms PALM-type baselines and PG in both iterations and runtime, with higher denoising quality as reflected by SNR gains. The approach thereby offers a practical and scalable framework for challenging nonconvex nonsmooth problems in applications.
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.