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

TL;DR

This paper develops convergence analysis for two-timescale simulations of hybrid systems combining continuous and discrete dynamics, with applications to stochastic hybrid optimization algorithms.

Contribution

It provides new conditions ensuring convergence of two-timescale simulations of hybrid inclusions and links stochastic approximations to deterministic hybrid systems.

Findings

Established sufficient conditions for convergence of two-timescale hybrid simulations.

Characterized limiting behavior via invariant and chain-transitive sets.

Demonstrated asymptotic recovery of deterministic behavior by stochastic hybrid algorithms.

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.