A Unified Algorithmic Framework of Symmetric Gauss-Seidel Decomposition Based Proximal ADMMs for Convex Composite Programming (Journal of Computational Mathematics, 2021)
This paper aims to present a fairly accessible generalization of several symmetric Gauss-Seidel decomposition based multi-block proximal alternating direction methods of multipliers (ADMMs) for convex composite optimization problems. The proposed method unifiesand refines many constructive techniques that were separately developed for the computational efficiency of multi-block ADMM-type algorithms. Specifically, the majorizedaugmented Lagrangian functions, the indefinite proximal terms, the inexact symmetricGauss-Seidel decomposition theorem, the tolerance criteria of approximately solving thesubproblems, and the large dual step-lengths, are all incorporated in one algorithmic framework, which we named as sGS-imiPADMM. From the popularity of convergent variantsof multi-block ADMMs in recent years, especially for high-dimensional multi-block convexcomposite conic programming problems, the unification presented in this paper, as wellas the corresponding convergence results, may have the great potential of facilitating theimplementation of many multi-block ADMMs in various problem settings.