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Distributed Adaptive Time-Varying Optimization with Global Asymptotic Convergence

Liangze Jiang, Zheng-Guang Wu, Lei Wang

TL;DR

This work tackles distributed time-varying optimization for multi-agent systems with dynamic costs by introducing a novel architecture that couples a dynamic average Hessian estimator with an adaptive optimizer, bridged by a Dead Zone Algorithm to handle singularities. The authors prove global asymptotic convergence to the time-varying optimizer $r^*(t)$ under mild assumptions, without requiring positive definite, diagonal, identical, or time-invariant Hessians, and without any waiting time in implementation. They extend the framework to general non-quadratic costs with corresponding estimators and provide corollaries, supported by simulations on multi-agent and UAV formation scenarios that demonstrate robustness to information loss and disturbances. The results offer a scalable, flexible approach for autonomous optimization in dynamic environments, with potential applications in large-scale robotic networks and sensor systems.

Abstract

In this note, we study distributed time-varying optimization for a multi-agent system. We first focus on a class of time-varying quadratic cost functions, and develop a new distributed algorithm that integrates an average estimator and an adaptive optimizer, with both bridged by a Dead Zone Algorithm. Based on a composite Lyapunov function and finite escape-time analysis, we prove the closed-loop global asymptotic convergence to the optimal solution under mild assumptions. Particularly, the introduction of the estimator relaxes the requirement for the Hessians of cost functions, and the integrated design eliminates the waiting time required in the relevant literature for estimating global parameter during algorithm implementation. We then extend this result to a more general class of time-varying cost functions. Two examples are used to verify the proposed designs.

Distributed Adaptive Time-Varying Optimization with Global Asymptotic Convergence

TL;DR

This work tackles distributed time-varying optimization for multi-agent systems with dynamic costs by introducing a novel architecture that couples a dynamic average Hessian estimator with an adaptive optimizer, bridged by a Dead Zone Algorithm to handle singularities. The authors prove global asymptotic convergence to the time-varying optimizer under mild assumptions, without requiring positive definite, diagonal, identical, or time-invariant Hessians, and without any waiting time in implementation. They extend the framework to general non-quadratic costs with corresponding estimators and provide corollaries, supported by simulations on multi-agent and UAV formation scenarios that demonstrate robustness to information loss and disturbances. The results offer a scalable, flexible approach for autonomous optimization in dynamic environments, with potential applications in large-scale robotic networks and sensor systems.

Abstract

In this note, we study distributed time-varying optimization for a multi-agent system. We first focus on a class of time-varying quadratic cost functions, and develop a new distributed algorithm that integrates an average estimator and an adaptive optimizer, with both bridged by a Dead Zone Algorithm. Based on a composite Lyapunov function and finite escape-time analysis, we prove the closed-loop global asymptotic convergence to the optimal solution under mild assumptions. Particularly, the introduction of the estimator relaxes the requirement for the Hessians of cost functions, and the integrated design eliminates the waiting time required in the relevant literature for estimating global parameter during algorithm implementation. We then extend this result to a more general class of time-varying cost functions. Two examples are used to verify the proposed designs.
Paper Structure (16 sections, 6 theorems, 66 equations, 7 figures, 1 algorithm)

This paper contains 16 sections, 6 theorems, 66 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Preliminaries
  4. Problem Definition
  5. Design Based on Estimator and Optimizer
  6. Average Estimator and Output Reconstruction
  7. Adaptive Optimizer
  8. Convergence Analysis
  9. Extensions: Non-Quadratic Cost Function
  10. Simulations
  11. Conclusion
  12. Two Auxiliary Lemmas
  13. Proof of Lemma \ref{['condition']}
  14. Proof of Theorem \ref{['distributed.oneorder']}
  15. Reanalysis of Finite Escape Time for Corollary \ref{['distributed.corollary']}
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose Assumptions assumption.graph1-assumption.boundedness.hessian hold. For system eq.z and Algorithm alg with $h_0=\gamma h_1/N$, there exist a constant ${k}_1>0$ such that $|\hat{z}_i(t)-\bar{H}(t)|\leq{k}_1|z_i(t)-\bar{H}(t)|$ holds $\forall t\geq 0$.

Figures (7)

  • Figure 1: Modeling of cost functions in dynamic environments.
  • Figure 2: The structure of the proposed design.
  • Figure 3: The interconnections within the proposed algorithm.
  • Figure 4: The trajectory of $|x_i-r^*(t)|$, $i\in\mathcal{N}$.
  • Figure 5: Multi-UAV relay communication. Purple ellipses denote robot working ranges. The devil icon denotes an attacker.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Remark 5
  • Corollary 1
  • ...and 7 more