Distributed continuous-time convex optimization on weight-balanced digraphs

TL;DR

This paper investigates continuous-time distributed convex optimization over directed graphs, demonstrating the failure of undirected dynamics in directed settings and proposing a new convergent dynamics for weight-balanced digraphs.

Contribution

It introduces a novel distributed dynamics that guarantees convergence on strongly connected weight-balanced digraphs, unlike previous consensus-based methods.

Findings

Existing dynamics fail on directed graphs

Proposed dynamics guarantees convergence

Applicable to strongly connected weight-balanced digraphs

Abstract

This paper studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.