Convergence in distribution of some particular self-interacting diffusions: the simulated annealing method

TL;DR

This paper investigates the convergence behavior of certain self-interacting diffusions used in simulated annealing, establishing conditions under which the process converges to the global minima of a potential function.

Contribution

It provides necessary and sufficient conditions for convergence of self-interacting diffusions in the context of simulated annealing, extending previous ergodic analysis.

Findings

Convergence in probability to the global minima under small g

Conditions linking the function g to convergence behavior

Analysis of ergodic properties of self-interacting diffusions

Abstract

The present paper is concerned with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V(X_t - \bar{\mu}_t) \mathrm{d}t$$ where $\bar{\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. The authors have still studied the ergodic behavior of $X$ and proved that it is strongly related to $g$. We go further and give necessary and sufficient conditions (for small $g$'s) in order that $X$ converges in probability to $X_\infty$ (which is related to the global minima of $V$).