# A Unified Algorithmic Framework of Symmetric Gauss-Seidel Decomposition based Proximal ADMMs for Convex Composite Programming

## TL;DR

This paper introduces a unified and generalized algorithmic framework called sGS-imiPADMM that combines various techniques for efficient multi-block convex composite optimization, improving convergence and implementation flexibility.

## Contribution

It unifies multiple existing techniques into a single framework for multi-block ADMMs, enhancing computational efficiency and convergence analysis.

## Key findings

- The proposed sGS-imiPADMM converges under broad conditions.
- The framework incorporates majorized augmented Lagrangian, indefinite proximal terms, and inexact solutions.
- Potential for improved implementation in high-dimensional convex conic programming.

## Abstract

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 unifies and refines many constructive techniques that were separately developed for the computational efficiency of multi-block ADMM-type algorithms. Specifically, the majorized augmented Lagrangian functions, the indefinite proximal terms, the inexact symmetric Gauss-Seidel decomposition theorem, the tolerance criteria of approximately solving the subproblems, and the large dual step-lengths, are all incorporated in one algorithmic framework, which we named as sGS-imiPADMM. From the popularity of convergent variants of multi-block ADMMs in recent years, especially for high-dimensional multi-block convex composite conic programming problems, the unification presented in this paper, as well as the corresponding convergence results, may have the great potential of facilitating the implementation of many multi-block ADMMs in various problem settings.