# Parallel Direction Method of Multipliers

arXiv: https://arxiv.org/abs/1406.4064 · HTML: https://mlsanity.com/paper/1406.4064/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2014.

## Built on by (most cited first, 2 of 2)

- [Personalized Federated Learning via Convex Clustering](https://mlsanity.com/paper/2202.00718/) (2022)
- [Transformed ℓ1 Regularizations for Robust Principal Component Analysis: Toward a Fine-Grained Understanding](https://mlsanity.com/paper/2510.03624/) (2025)

## Abstract (arXiv)

We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates some primal and dual blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso.