# A New Randomized Block-Coordinate Primal-Dual Proximal Algorithm for Distributed Optimization

## TL;DR

This paper introduces TriPD, a primal-dual algorithm with a randomized block-coordinate variant for distributed optimization, achieving linear convergence under certain conditions and enabling fully distributed multi-agent applications.

## Contribution

The paper presents a novel randomized block-coordinate primal-dual algorithm with proven convergence and applies it to distributed multi-agent optimization without global coordination.

## Key findings

- Converges under the same stepsize conditions as the full algorithm.
- Achieves linear convergence for piecewise linear-quadratic functions.
- Enables fully distributed multi-agent optimization with local information.

## Abstract

This paper proposes TriPD, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the block-coordinate scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is weaker than strong convexity). Moreover, we apply the developed algorithms to the problem of multi-agent optimization on a graph, thus obtaining novel synchronous and asynchronous distributed methods. The proposed algorithms are fully distributed in the sense that the updates and the stepsizes of each agent only depend on local information. In fact, no prior global coordination is required. Finally, we showcase an application of our algorithm in distributed formation control.