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Event-triggered Dual Gradient Tracking for Distributed Resource Allocation

Xiayan Xu, Xiaomeng Chen, Dawei Shi, Ling Shi

TL;DR

The paper tackles high communication costs in distributed resource allocation over unbalanced directed graphs by introducing an event-triggered dual gradient tracking (ET-DGT) framework built on push-pull gradient tracking. It develops a general event-triggered push-pull gradient (ET-PPG) theory, proving sublinear convergence for nonconvex objectives and linear convergence under the Polyak-Łojasiewicz condition, with analogous results for strongly convex and Lipschitz-smooth costs. The ET-DGT algorithm achieves significant communication reductions while preserving convergence rates comparable to periodic schemes, as demonstrated on IEEE 14- and 118-bus network simulations, including scenarios with non-quadratic costs. The findings offer a scalable, communication-efficient approach for distributed resource allocation in directed networks, with potential applicability to smart grids and similar large-scale, constrained multi-agent systems.

Abstract

High communication costs create a major bottleneck for distributed resource allocation over unbalanced directed networks. Conventional dual gradient tracking methods, while effective for problems on unbalanced digraphs, rely on periodic communication that creates significant overhead in resource-constrained networks. This paper introduces a novel event-triggered dual gradient tracking algorithm to mitigate this limitation, wherein agents communicate only when local state deviations surpass a predefined threshold. We establish comprehensive convergence guarantees for this approach. First, we prove sublinear convergence for non-convex dual objectives and linear convergence under the Polyak-Łojasiewicz condition. Building on this, we demonstrate that the proposed algorithm achieves sublinear convergence for general strongly convex cost functions and linear convergence for those that are also Lipschitz-smooth. Numerical experiments confirm that our event-triggered method significantly reduces communication events compared to periodic schemes while preserving comparable convergence performance.

Event-triggered Dual Gradient Tracking for Distributed Resource Allocation

TL;DR

The paper tackles high communication costs in distributed resource allocation over unbalanced directed graphs by introducing an event-triggered dual gradient tracking (ET-DGT) framework built on push-pull gradient tracking. It develops a general event-triggered push-pull gradient (ET-PPG) theory, proving sublinear convergence for nonconvex objectives and linear convergence under the Polyak-Łojasiewicz condition, with analogous results for strongly convex and Lipschitz-smooth costs. The ET-DGT algorithm achieves significant communication reductions while preserving convergence rates comparable to periodic schemes, as demonstrated on IEEE 14- and 118-bus network simulations, including scenarios with non-quadratic costs. The findings offer a scalable, communication-efficient approach for distributed resource allocation in directed networks, with potential applicability to smart grids and similar large-scale, constrained multi-agent systems.

Abstract

High communication costs create a major bottleneck for distributed resource allocation over unbalanced directed networks. Conventional dual gradient tracking methods, while effective for problems on unbalanced digraphs, rely on periodic communication that creates significant overhead in resource-constrained networks. This paper introduces a novel event-triggered dual gradient tracking algorithm to mitigate this limitation, wherein agents communicate only when local state deviations surpass a predefined threshold. We establish comprehensive convergence guarantees for this approach. First, we prove sublinear convergence for non-convex dual objectives and linear convergence under the Polyak-Łojasiewicz condition. Building on this, we demonstrate that the proposed algorithm achieves sublinear convergence for general strongly convex cost functions and linear convergence for those that are also Lipschitz-smooth. Numerical experiments confirm that our event-triggered method significantly reduces communication events compared to periodic schemes while preserving comparable convergence performance.

Paper Structure

This paper contains 28 sections, 13 theorems, 103 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Organization of the Paper
  5. Problem Statement
  6. Distributed Resource Allocation Problem
  7. Dual Problem of DRA
  8. Distributed Dual Gradient Tracking Algorithm
  9. Event-triggered Algorithm Development
  10. Event-triggered Mechanisms
  11. Event-triggered Dual Gradient Tracking Algorithm
  12. Convergence Analysis
  13. Supporting Lemmas
  14. Convergence of Dual Problem
  15. Convergence of Primal Problem
  16. ...and 13 more sections

Key Result

Lemma 1

The row-stochastic matrix $\mathbf{R}$ admits a unique unit non-negative left eigenvector $\pi_{\mathbf{R}}$ w.r.t. eigenvalue $1$, i.e., $\pi_{\mathbf{R}}^{\top}\mathbf{R}=\pi_{\mathbf{R}}^{\top}$ and $\pi_{\mathbf{R}}^{\top}\mathbf{1}=1$. The column-stochastic matrix $\mathbf{C}$ admits a unique u

Figures (5)

  • Figure 1: Convergence of individual generator outputs under the proposed ET-DGT algorithm in Case 1.
  • Figure 2: Total power generation v.s. total demand for all compared algorithms in Case 1.
  • Figure 3: Convergence error comparison among algorithms in Case 1.
  • Figure 4: Convergence error comparison among algorithms in Case 2.
  • Figure 5: Convergence error comparison among algorithms in Case 3.

Theorems & Definitions (22)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 12 more