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

## TL;DR

This paper introduces a novel derivation of the primal-dual method of multipliers (PDMM) using monotone operator theory, connecting it with other first-order methods and providing convergence insights for distributed optimization.

## Contribution

It presents a new derivation of PDMM via monotone operator theory, linking it to Douglas-Rachford splitting and establishing convergence conditions.

## Key findings

- PDMM can be derived using monotone operator theory.
- Sufficient conditions for strong primal convergence are provided.
- A distributed parameter selection method with finite transmissions is introduced.

## Abstract

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.