# Convergence Revisit on Generalized Symmetric ADMM

## TL;DR

This paper revisits the convergence properties of the generalized symmetric ADMM algorithm, establishing sublinear and linear convergence rates under specific conditions, thereby enhancing understanding of its theoretical performance.

## Contribution

It provides new convergence rate results for the generalized symmetric ADMM, including sublinear and linear rates under particular assumptions and parameter settings.

## Key findings

- Sublinear nonergodic convergence rate established.
- Linear convergence under piecewise linear sub-differential and polyhedral constraints.
- Convergence results depend on dual stepsize parameters within a specific isosceles triangle region.

## Abstract

In this note, we show a sublinear nonergodic convergence rate for the algorithm developed in [Bai, et al. Generalized symmetric ADMM for separable convex optimization. Comput. Optim. Appl. 70, 129-170 (2018)], as well as its linear convergence under assumptions that the sub-differential of each component objective function is piecewise linear and all the constraint sets are polyhedra. These remaining convergence results are established for the stepsize parameters of dual variables belonging to a special isosceles triangle region, which aims to strengthen our understanding for convergence of the generalized symmetric ADMM.