Simulated Annealing In $\mathbf{R}^d$ With Slowly Growing Potentials

TL;DR

This paper investigates the behavior of continuous-time simulated annealing in high-dimensional spaces with slowly growing potentials, identifying a phase transition at potentials growing like a double logarithm.

Contribution

It weakens previous growth assumptions for potentials in simulated annealing and characterizes the transition point for potentials growing like a double logarithm.

Findings

Transition occurs at potentials growing like a log log |x|

Applicable to potentials with unbounded local minima

Extends previous results with weaker growth conditions

Abstract

We use a localization procedure to weaken the growth assumptions of Royer [8], Miclo [4] and Zitt [9] concerning the continuous-time simulated annealing in $\mathbf{R}^d$. We show that a transition occurs for potentials growing like $a \log \log |x|$ at infinity. We also study a class of potentials with possibly unbounded sets of local minima.