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Time-delay Induced Stochastic Optimization and Extremum Seeking

Naum Dimitrieski, Michael Reyer, Mohamed-Ali Belabbas, Christian Ebenbauer

Abstract

In this paper a novel stochastic optimization and extremum seeking algorithm is presented, one which is based on time-delayed random perturbations and step size adaptation. For the case of a one-dimensional quadratic unconstrained optimization problem, global exponential convergence in expectation and global exponential practical convergence of the variance of the trajectories are proven. The theoretical results are complemented by numerical simulations for one- and multi-dimensional quadratic and non-quadratic objective functions.

Time-delay Induced Stochastic Optimization and Extremum Seeking

Abstract

In this paper a novel stochastic optimization and extremum seeking algorithm is presented, one which is based on time-delayed random perturbations and step size adaptation. For the case of a one-dimensional quadratic unconstrained optimization problem, global exponential convergence in expectation and global exponential practical convergence of the variance of the trajectories are proven. The theoretical results are complemented by numerical simulations for one- and multi-dimensional quadratic and non-quadratic objective functions.

Paper Structure

This paper contains 7 sections, 12 theorems, 139 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and Main Result
  3. Simulations
  4. Conclusions and Outlook
  5. Appendix
  6. Auxiliary Results
  7. Proof of Main Result

Key Result

Theorem 1

Consider system sys_def with assump:ghw, suppose assump:quadratic holds, and let $\chi, \psi, \mu \in \mathbb{R}^{+}$ be arbitrary. Assume that ${\rho, \beta,\varepsilon}$ satisfy where ${ \rho_0, \beta_0, \varepsilon_0 \in \mathbb{R}^{+}}$ are sufficiently small. Then, for any $x_0,y_0 \in \mathbb{R}$ the system trajectory ${(x_k, y_k)}$ converges exponentially in expectation to $(x^{*},0)$ and

Figures (10)

  • Figure 1: Simulation result for system \ref{['sys_def']} with objective function \ref{['one_dim_q']}
  • Figure 2: Standard deviation of system \ref{['sys_def']} with the objective function \ref{['one_dim_q']}
  • Figure 3: Sample trajectories of the $x-$subsystem in \ref{['sys_def']} with objective function \ref{['one_dim_q']}
  • Figure 4: Sample trajectories of the $y-$subsystem in \ref{['sys_def']} with objective function \ref{['one_dim_q']}
  • Figure 5: Simulation result for system \ref{['sys_def']} with objective function \ref{['one_dim_q']}
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 1
  • Proposition 1
  • proof : Proof of \ref{['lemma:matrix_upper_lower_bounds']}
  • lemma 1
  • proof : Proof of \ref{['proposition:quadratic_decomposition']}
  • lemma 2
  • proof : Proof of \ref{['proposition:general_properties']}
  • Remark 1
  • lemma 3
  • proof : Proof of \ref{['proposition:expectation_h_k-1_delta_k-1']}
  • ...and 15 more