# Derivation and Analysis of the Primal-Dual Method of Multipliers Based on Monotone Operator Theory

arXiv: https://arxiv.org/abs/1706.02654 · HTML: https://mlsanity.com/paper/1706.02654/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2017.

## Built on by (most cited first, 9 of 9)

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- [Communication Compression for Decentralized Learning with Operator Splitting Methods](https://mlsanity.com/paper/2205.03779/) (2022)
- [Limited Communications Distributed Optimization via Deep Unfolded Distributed ADMM](https://mlsanity.com/paper/2309.14353/) (2023)
- [OpReg-Boost: Learning to Accelerate Online Algorithms with Operator Regression](https://mlsanity.com/paper/2105.13271/) (2021)
- [Theoretical Analysis of Primal-Dual Algorithm for Non-Convex Stochastic Decentralized Optimization](https://mlsanity.com/paper/2205.11979/) (2022)
- [AttentionX: Exploiting Consensus Discrepancy In Attention from A Distributed Optimization Perspective](https://mlsanity.com/paper/2409.04275/) (2024)

## Abstract (arXiv)

In this paper we present a novel derivation for an existing node-based algorithm for distributed optimisation termed the primal-dual method of multipliers (PDMM). In contrast to its initial derivation, in this work monotone operator theory is used to connect PDMM with other first-order methods such as Douglas-Rachford splitting and the alternating direction method of multipliers thus providing insight to the operation of the scheme. In particular, we show how PDMM combines a lifted dual form in conjunction with Peaceman-Rachford splitting to remove the need for collaboration between nodes per iteration. We demonstrate sufficient conditions for strong primal convergence for a general class of functions while under the assumption of strong convexity and functional smoothness, we also introduce a primal geometric convergence bound. Finally we introduce a distributed method of parameter selection in the geometric convergent case, requiring only finite transmissions to implement regardless of network topology.