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Delay-Robust Primal-Dual Dynamics for Distributed Optimization

Gökçen Devlet Şen, Juan E. Machado, Gülay Öke Günel, Johannes Schiffer

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.

Delay-Robust Primal-Dual Dynamics for Distributed Optimization

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.
Paper Structure (10 sections, 4 theorems, 28 equations, 1 table)

This paper contains 10 sections, 4 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. An Augmented Primal-Dual Gradient Dynamics for Enhanced Delay-Robustness
  4. Augmented PDGD
  5. Gain Synthesis with Guaranteed Delay-Robust Stability
  6. State-Independent Reformulation
  7. Numerical Example
  8. Conclusion
  9. Proof of Proposition \ref{['prop:stability_cond']}
  10. Proof of Corollary \ref{['corollary:LMI_formulation']}

Key Result

Lemma III.1

The point $(\bar{x}, \bar{\lambda}, \bar{u}^1, \bar{u}^2)$ is an equilibrium of eq:proposed_dyn if and only if the pair $(\bar{x}, \bar{\lambda})$ satisfies the KKT conditions eq:KKT_cond.

Theorems & Definitions (8)

  • Lemma III.1
  • proof
  • Lemma III.2: Positive definiteness qu2018exponential
  • Proposition III.1
  • Corollary III.1
  • Remark 1
  • Remark 2
  • Remark 3