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Two-Timescale Asymptotic Simulations of Hybrid Inclusions with Applications to Stochastic Hybrid Optimization

Max F. Crisafulli, Andrew R. Teel

Abstract

Convergence properties of model-free two-timescale asymptotic simulations of singularly perturbed hybrid inclusions are developed. A hybrid inclusion combines constrained differential and difference inclusions to capture continuous (flow) and discrete (jump) dynamics, respectively. Sufficient conditions are established under which sequences of iterates and step sizes constitute a two-timescale asymptotic simulation of such a system, with limiting behavior characterized via weakly invariant and internally chain-transitive sets of an associated boundary layer and reduced system. To illustrate the applicability of these results, conditions are given under which a two-timescale stochastic approximation of a hybrid optimization algorithm asymptotically recovers the behavior of its deterministic counterpart.

Two-Timescale Asymptotic Simulations of Hybrid Inclusions with Applications to Stochastic Hybrid Optimization

Abstract

Convergence properties of model-free two-timescale asymptotic simulations of singularly perturbed hybrid inclusions are developed. A hybrid inclusion combines constrained differential and difference inclusions to capture continuous (flow) and discrete (jump) dynamics, respectively. Sufficient conditions are established under which sequences of iterates and step sizes constitute a two-timescale asymptotic simulation of such a system, with limiting behavior characterized via weakly invariant and internally chain-transitive sets of an associated boundary layer and reduced system. To illustrate the applicability of these results, conditions are given under which a two-timescale stochastic approximation of a hybrid optimization algorithm asymptotically recovers the behavior of its deterministic counterpart.

Paper Structure

This paper contains 13 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Hybrid Inclusions
  3. Weak Invariance and Internal Chain Transitivity
  4. Asymptotic Simulations of Hybrid Inclusions
  5. Hybrid Sequences
  6. Asymptotic Simulations
  7. Two-Timescale Asymptotic Simulations
  8. Two-Timescale System
  9. Boundary Layer System
  10. Reduced System
  11. Example - Stochastic Hybrid Optimization
  12. Hybrid Heavy Ball Algorithm
  13. Two-Timescale Stochastic Approximation

Key Result

Proposition 1

Let the system $\Hcal$ satisfy Assumption ass:basic_conditions and $(\phi,\{h_k\}_{k=1}^\infty)$ be an asymptotic simulation of $\Hcal$. Then, $\omega(\phi)$ is a nonempty and compact set that is both weakly $\mathcal{S}_\mathcal{H}$-invariant and internally $\mathcal{S}_\mathcal{H}$-chain transitiv

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • Definition 6
  • Definition 7
  • Remark 1
  • Theorem 1
  • ...and 18 more