Delay-Robust Primal-Dual Dynamics for Distributed Optimization

TL;DR

This paper introduces a delay-robust continuous-time primal-dual gradient dynamics for distributed optimization, enhancing stability and performance under communication delays through auxiliary states and specific gain tuning.

Contribution

It proposes a novel delay-robust primal-dual dynamics with a gain matrix, along with linear matrix inequality conditions for stability under delays.

Findings

Improved delay robustness over standard PDGD.

Sufficient conditions for stability via LMIs.

Numerical example demonstrating enhanced performance.

Abstract

Continuous-time primal-dual gradient dynamics (PDGD) is an ubiquitous approach for dynamically solving constrained distributed optimization problems. Yet, the distributed nature of the dynamics makes it prone to communication uncertainties, especially time delays. To mitigate this effect, we propose a delay-robust continuous-time PDGD. The dynamics is obtained by augmenting the standard PDGD with an auxiliary state coupled through a gain matrix, while preserving the optimal solution. Then, we present sufficient tuning conditions for this gain matrix in the form of linear matrix inequalities, which ensure uniform asymptotic stability in the presence of bounded, time-varying delays. The criterion is derived via the Lyapunov-Krasovskii method. A numerical example illustrates the improved delay robustness of our approach compared to the standard PDGD under large, time-varying delays.