On The Simulated Annealing In $\mathbf{R}^d$

TL;DR

This paper extends the theoretical understanding of continuous-time simulated annealing in high-dimensional spaces by weakening growth assumptions and establishing conditions for invariant measures, with additional results on diffusion non-explosion.

Contribution

It significantly broadens the conditions under which simulated annealing is proven to succeed in , relying only on the existence of an invariant measure at low temperature.

Findings

Weakened growth assumptions for simulated annealing success.

Proved existence of invariant probability measures at low temperature.

Provided a non-explosion criterion for certain diffusions.

Abstract

Using a localization procedure and the result of Holley-Kusuoka-Stroock [7] in the torus, we widely weaken the usual growth assumptions concerning the success of the continuous-time simulated annealing in $\mathbf{R}^d$. Our only assumption is the existence of an invariant probability measure for a sufficiently low temperature. We also prove, in an appendix, a non-explosion criterion for a class of time-inhomogeneous diffusions.