Fast simulated annealing in $\R^d$ and an application to maximum   likelihood estimation
  Using classical simulated annealing to maximise a function $\psi$ defined on a subset of $\R^d$, the probability $\p(\psi(\theta\_n)\leq \psi\_{\max}-\epsilon)$ tends to zero at a logarithmic rate as $n$ increases; here $\theta\_n$ is the state in the $n$-th stage of the simulated annealing algorithm and $\psi\_{\max}$ is the maximal value of $\psi$. We propose a modified scheme for which this probability is of order $n^{-1/3}\log n$, and hence vanishes at an algebraic rate. To obtain this faster rate, the exponentially decaying acceptance probability of classical simulated annealing is replaced by a more heavy-tailed function, and the system is cooled faster. 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. 
Sylvain Rubenthaler (JAD), Tobias Ryd\'en (CENTRE for Mathematical   Sciences), Magnus Wiktorsson (CENTRE for Mathematical S...