Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An online optimization algorithm for tracking a linearly varying optimal point with zero steady-state error

Alex Xinting Wu, Ian R. Petersen, Valery Ugrinovskii, Iman Shames

TL;DR

The paper tackles online optimization where the optimal point drifts linearly in time and the gradient lies in a sector, formulating the update as a Luré-type system with a double integrator to ensure ramp-tracking with zero steady-state error. It leverages the discrete-time circle criterion to guarantee absolute stability and derives an explicit R-convergence bound, identifying optimal gains $\alpha^*=\frac{2}{L}$ and $\gamma^*=\frac{1}{m+L}$ that yield $\rho^*(\kappa)=\sqrt{(\kappa-1)/(\kappa+1)}$ with $\kappa=L/m$. The results show that no circle-criterion-based method can outperform this bound for the given $\kappa$, and the TOA localization example confirms zero steady-state error in tracking a linearly moving source despite transients. Overall, the work provides a principled online optimization algorithm with provable zero steady-state tracking error and a concrete performance metric for nonconvex, time-varying problems.

Abstract

In this paper, we develop an online optimization algorithm for solving a class of nonconvex optimization problems with a linearly varying optimal point. The global convergence of the algorithm is guaranteed using the circle criterion for the class of functions whose gradient is bounded within a sector. Also, we show that the corresponding Luré-type nonlinear system involves a double integrator, which demonstrates its ability to track a linearly varying optimal point with zero steady-state error. The algorithm is applied to solving a time-of-arrival based localization problem with constant velocity and the results show that the algorithm is able to estimate the source location with zero steady-state error.

An online optimization algorithm for tracking a linearly varying optimal point with zero steady-state error

TL;DR

The paper tackles online optimization where the optimal point drifts linearly in time and the gradient lies in a sector, formulating the update as a Luré-type system with a double integrator to ensure ramp-tracking with zero steady-state error. It leverages the discrete-time circle criterion to guarantee absolute stability and derives an explicit R-convergence bound, identifying optimal gains and that yield with . The results show that no circle-criterion-based method can outperform this bound for the given , and the TOA localization example confirms zero steady-state error in tracking a linearly moving source despite transients. Overall, the work provides a principled online optimization algorithm with provable zero steady-state tracking error and a concrete performance metric for nonconvex, time-varying problems.

Abstract

In this paper, we develop an online optimization algorithm for solving a class of nonconvex optimization problems with a linearly varying optimal point. The global convergence of the algorithm is guaranteed using the circle criterion for the class of functions whose gradient is bounded within a sector. Also, we show that the corresponding Luré-type nonlinear system involves a double integrator, which demonstrates its ability to track a linearly varying optimal point with zero steady-state error. The algorithm is applied to solving a time-of-arrival based localization problem with constant velocity and the results show that the algorithm is able to estimate the source location with zero steady-state error.

Paper Structure

This paper contains 8 sections, 6 theorems, 82 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. Background and Preliminary Results
  3. Heavy Ball Method
  4. Generalized accelerated gradient method
  5. Preliminary Results
  6. Main Result
  7. Illustrative Example
  8. CONCLUSIONS

Key Result

Lemma II.1

Given any $\alpha$, $\gamma$, $m$, $L$, such that the algorithm $\Sigma(\alpha,\gamma,m,L)$ is globally asymptotically convergent, where $\rho(\cdot)$ denote the spectral radius of its matrix argument.

Figures (7)

  • Figure 1: Block diagram corresponding to Luré system representation
  • Figure 2: Triangle shaped region $T_\rho$.
  • Figure 3: Illustration of Case 1.
  • Figure 4: Illustration of Case 2.
  • Figure 5: Comparison between the algorithm (\ref{['uprulebeta1']}), the gradient descent algorithm Boyd_Vandenberghe_2004 and the triple momentum method 7967721 applied to the TOA-based localization problem (\ref{['LQTOA']}), with $L = 6$ and $m = 0.1$, to estimate the true source location $x_1$ at different iteration steps $t$.
  • ...and 2 more figures

Theorems & Definitions (16)

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