# Distributed Optimization Using the Primal-Dual Method of Multipliers

arXiv: https://arxiv.org/abs/1702.00841 · HTML: https://mlsanity.com/paper/1702.00841/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2017.

## Built on by (most cited first, 13 of 13)

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- [Communication Compression for Decentralized Learning with Operator Splitting Methods](https://mlsanity.com/paper/2205.03779/) (2022)
- [Privacy-Preserving Distributed Expectation Maximization for Gaussian Mixture Model using Subspace Perturbation](https://mlsanity.com/paper/2209.07833/) (2022)
- [Limited Communications Distributed Optimization via Deep Unfolded Distributed ADMM](https://mlsanity.com/paper/2309.14353/) (2023)
- [Theoretical Analysis of Primal-Dual Algorithm for Non-Convex Stochastic Decentralized Optimization](https://mlsanity.com/paper/2205.11979/) (2022)
- [Lightweight Trustworthy Distributed Clustering](https://mlsanity.com/paper/2504.10109/) (2025)
- [LAGO: Few-shot Crosslingual Embedding Inversion Attacks via Language Similarity-Aware Graph Optimization](https://mlsanity.com/paper/2505.16008/) (2025)
- [AttentionX: Exploiting Consensus Discrepancy In Attention from A Distributed Optimization Perspective](https://mlsanity.com/paper/2409.04275/) (2024)
- [Resource-Efficient Distributed Recursive Gaussian Processes](https://mlsanity.com/paper/2609.26979/) (2026)

## Abstract (arXiv)

In this paper, we propose the primal-dual method of multipliers (PDMM) for distributed optimization over a graph. In particular, we optimize a sum of convex functions defined over a graph, where every edge in the graph carries a linear equality constraint. In designing the new algorithm, an augmented primal-dual Lagrangian function is constructed which smoothly captures the graph topology. It is shown that a saddle point of the constructed function provides an optimal solution of the original problem. Further under both the synchronous and asynchronous updating schemes, PDMM has the convergence rate of O(1/K) (where K denotes the iteration index) for general closed, proper and convex functions. Other properties of PDMM such as convergence speeds versus different parameter- settings and resilience to transmission failure are also investigated through the experiments of distributed averaging.