Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization

2025/10/31 by Xian-Jun Long, Long, Xian-Jun, Kai Zeng +7
Engineering · Computer Science · #Sparse and Compressive Sensing Techniques #Image and Signal Denoising Methods #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.2510.27156

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 ε-approximate stationary point is O(ε^-3). The proposed algorithm is evaluated on sparse signal recovery and image denoising problems. Numerical experiments demonstrate its effectiveness and superiority over existing algorithms.

Citations

• Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex Optimization with Applications to Image Restoration
• iPiano: Inertial Proximal Algorithm for Non-Convex Optimization
• Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization
• Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization
• Proximal alternating linearized minimization for nonconvex and nonsmooth problems
• Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods
• A variable smoothing algorithm for solving convex optimization problems
• A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging
• Nearly unbiased variable selection under minimax concave penalty
• Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka-Lojasiewicz inequality
• Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

Related

• A Proximal Variable Smoothing for Minimization of Nonlinearly Composite Nonsmooth Function -- Finite-Max Minimization and MIMO Applications
• Smoothing technique for nonsmooth composite minimization with linear operator
• Smoothing Accelerated Proximal Gradient Method with Backtracking for Nonsmooth Multiobjective Optimization
• Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization
• The proximal-proximal gradient algorithm