# Distributed Optimization Using the Primal-Dual Method of Multipliers

## TL;DR

This paper introduces PDMM, a novel primal-dual algorithm for distributed convex optimization over graphs, demonstrating convergence and robustness under various update schemes and network conditions.

## Contribution

The paper develops PDMM, a new distributed optimization method that effectively handles graph-structured problems with convergence guarantees and resilience to communication failures.

## Key findings

- Converges at rate O(1/K) for convex functions.
- Effective under both synchronous and asynchronous updates.
- Resilient to transmission failures in distributed averaging.

## Abstract

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.