Fast simulated annealing in R-d with an application to maximum likelihood estimation in state-space models

Abstract:
We study simulated annealing algorithms to maximise a function psi on a subset of R-d. In classical simulated annealing, given a current state theta(n) in stage n of the algorithm, the probability to accept a proposed state z at which psi is smaller, is exp(-beta(n+1)(psi(z) - psi (theta(n))) where (beta(n)) is the inverse temperature. With the standard logarithmic increase of (beta(n)) the probability P(psi(theta(n)) <= psi(max) - epsilon), with psi(max) the maximal value of psi, then tends to zero at a logarithmic rate as n increases. We examine variations of this scheme in which (beta(n)) is allowed to grow faster, but also consider other functions than the exponential for determining acceptance probabilities. The main result shows that faster rates of convergence can be obtained, both with the exponential and other acceptance functions. We also show how the algorithm may be applied to functions that cannot be computed exactly but only approximated, and give an example of maximising the log-likelihood function for a state-space model. (C) 2008 Elsevier B.V. All rights reserved.