# Faster convergence rates of relaxed Peaceman-Rachford and ADMM under regularity assumptions

arXiv: https://arxiv.org/abs/1407.5210 · HTML: https://mlsanity.com/paper/1407.5210/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2014.

## Built on by (most cited first, 4 of 4)

- [Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions](https://mlsanity.com/paper/2110.08150/) (2021)
- [Complexity of Linear Minimization and Projection on Some Sets](https://mlsanity.com/paper/2101.10040/) (2021)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Operator Splitting for Convex Constrained Markov Decision Processes](https://mlsanity.com/paper/2412.14002/) (2024)

## Abstract (arXiv)

Splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which the simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable algorithms, which often obtain nearly state-of-the-art performance. In this paper, we provide a comprehensive convergence rate analysis of the Douglas-Rachford splitting (DRS), Peaceman-Rachford splitting (PRS), and alternating direction method of multipliers (ADMM) algorithms under various regularity assumptions including strong convexity, Lipschitz differentiability, and bounded linear regularity. The main consequence of this work is that relaxed PRS and ADMM automatically adapt to the regularity of the problem and achieve convergence rates that improve upon the (tight) worst-case rates that hold in the absence of such regularity. All of the results are obtained using simple techniques.