A primal-dual splitting algorithm with convex combination and larger step sizes for composite monotone inclusion problems
Xiaokai Chang, Junfeng Yang, Jianchao Bai, Jianxiong Cao
TL;DR
The paper addresses solving composite monotone inclusion problems of the form $0\in A(x)+K^{*}B(Kx)$ by developing a primal–dual splitting algorithm that blends extrapolation with a convex combination step. By reformulating the method as a fixed‑point iteration of an extended firmly nonexpansive operator $T_P$ under the condition $0\prec P\prec \Phi_K$, the authors prove convergence under a relaxed step‑size bound $\gamma\|K\|^{2}<(2-\theta)(2-\eta)$ and establish $O(1/N)$ ergodic rates for the associated convex problems. They also demonstrate the algorithm’s practical benefits—able to employ larger step sizes than CP‑PDHG and with favorable performance in imaging tasks, matrix games, and LASSO—alongside a discussion of sharpness and adaptive parameter strategies. The work thus provides a robust, full‑splitting framework for CMIPs with a flexible parameter regime and solid convergence guarantees, supported by extensive numerical experiments. Overall, the proposed method advances first‑order splitting techniques by enabling larger steps and offering fixed‑point‑theory–driven convergence analysis for a broad class of monotone inclusions.
Abstract
The primal-dual splitting algorithm (PDSA) by Chambolle and Pock is efficient for solving structured convex optimization problems. It adopts an extrapolation step and achieves convergence under certain step size condition. Chang and Yang recently proposed a modified PDSA for bilinear saddle point problems, integrating a convex combination step to enable convergence with extended step sizes. In this paper, we focus on composite monotone inclusion problems (CMIPs), a generalization of convex optimization problems. While Vu extended PDSA to CMIPs, whether the modified PDSA can be directly adapted to CMIPs remains an open question. This paper introduces a new PDSA for CMIPs, featuring the inclusion of both an extrapolation step and a convex combination step. The proposed algorithm is reformulated as a fixed-point iteration by leveraging an extended firmly nonexpansive operator. Under a significantly relaxed step size condition, both its convergence and sublinear convergence rate results are rigorously established. For structured convex optimization problem, we establish its sublinear convergence rate results measured by function value gap and constraint violations. Moreover, we show through a concrete example that our condition on the involved parameters cannot be relaxed. Numerical experiments on image denoising, inpainting, matrix games, and LASSO problems are conducted to compare the proposed algorithm with state-of-the-art counterparts, demonstrating the efficiency of the proposed algorithm.