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Distributed Continuous-Time Optimization with Uncertain Time-Varying Quadratic Cost Functions

Liangze Jiang, Zheng-Guang Wu, Lei Wang

TL;DR

The paper tackles distributed continuous-time optimization with time-varying quadratic costs containing uncertain parameters. It develops a centralized adaptive controller and two cascaded distributed schemes to achieve exact optimization despite uncertainty, including both identical and nonidentical Hessian cases; the key innovation is the use of state-based gains and a fixed-time average estimator to remove reliance on upper bounds and ensure global convergence. The contributions broaden the applicability of distributed optimization to uncertain, dynamic environments and are validated through simulations on moving-source and multi-robot scenarios, demonstrating robust tracking of the time-varying minimizer $x^*(t)$. Overall, the work offers theoretically sound, practically relevant designs for coordinated optimization in unknown, time-varying settings with convergent guarantees.

Abstract

This paper studies distributed continuous-time optimization for time-varying quadratic cost functions with uncertain parameters. We first propose a centralized adaptive optimization algorithm using partial information of the cost function. It can be seen that even if there are uncertain parameters in the cost function, exact optimization can still be achieved. To solve this problem in a distributed manner when different local cost functions have identical Hessians, we propose a novel distributed algorithm that cascades the fixed-time average estimator and the distributed optimizer. We remove the requirement for the upper bounds of certain complex functions by integrating state-based gains in the proposed design. We further extend this result to address the distributed optimization where the time-varying cost functions have nonidentical Hessians. We prove the convergence of all the proposed algorithms in the global sense. Numerical examples verify the proposed algorithms.

Distributed Continuous-Time Optimization with Uncertain Time-Varying Quadratic Cost Functions

TL;DR

The paper tackles distributed continuous-time optimization with time-varying quadratic costs containing uncertain parameters. It develops a centralized adaptive controller and two cascaded distributed schemes to achieve exact optimization despite uncertainty, including both identical and nonidentical Hessian cases; the key innovation is the use of state-based gains and a fixed-time average estimator to remove reliance on upper bounds and ensure global convergence. The contributions broaden the applicability of distributed optimization to uncertain, dynamic environments and are validated through simulations on moving-source and multi-robot scenarios, demonstrating robust tracking of the time-varying minimizer . Overall, the work offers theoretically sound, practically relevant designs for coordinated optimization in unknown, time-varying settings with convergent guarantees.

Abstract

This paper studies distributed continuous-time optimization for time-varying quadratic cost functions with uncertain parameters. We first propose a centralized adaptive optimization algorithm using partial information of the cost function. It can be seen that even if there are uncertain parameters in the cost function, exact optimization can still be achieved. To solve this problem in a distributed manner when different local cost functions have identical Hessians, we propose a novel distributed algorithm that cascades the fixed-time average estimator and the distributed optimizer. We remove the requirement for the upper bounds of certain complex functions by integrating state-based gains in the proposed design. We further extend this result to address the distributed optimization where the time-varying cost functions have nonidentical Hessians. We prove the convergence of all the proposed algorithms in the global sense. Numerical examples verify the proposed algorithms.
Paper Structure (16 sections, 6 theorems, 55 equations, 10 figures)

This paper contains 16 sections, 6 theorems, 55 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Preliminaries
  4. Problem Definition
  5. Design From a Centralized Perspective
  6. Design From a Distributed Perspective
  7. Design for cost Functions With Identical Hessians
  8. Design for cost Functions With Nonidentical Hessians
  9. Simulation
  10. Case 1
  11. Case 2
  12. Case 3
  13. Conclusion
  14. Proof of Lemma \ref{['lemma.finiteconverge']}
  15. Proof of Theorem \ref{['distributed.oneorder']}
  16. ...and 1 more sections

Key Result

Lemma 1

Polyakov12 Consider the system $\dot{x}=g(x,t)$, $x(0)=x_0$, where $x\in\mathbb{R}^m$ and $g:\mathbb{R}^n\times\mathbb{R}_+\rightarrow\mathbb{R}^m$ is a nonlinear function. Suppose there exists a continuous function $V:\mathbb{R}^m\rightarrow\mathbb{R}$ and some positive constants $a,b>0$ and $0<\mu

Figures (10)

  • Figure 1: The block diagram of the closed-loop system.
  • Figure 2: The trajectories of $y^*(t)$ and $y$.
  • Figure 3: The trajectories of $||\hat{\eta}_1||$ and $||y-y^*(t)||$.
  • Figure 4: Information exchange topology among robots.
  • Figure 5: The trajectory of $||(M\otimes I_2) \xi||$.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4
  • Lemma 3
  • proof
  • ...and 6 more