Computing the initial temperature of simulated annealing

The classical version of simulated annealing is based on a cooling schedule. Generally, the initial temperature is set such that the acceptance ratio of bad moves is equal to a certain value chi(0). In this paper, we first propose a simple algorithm to compute a temperature which is compatible with a given acceptance ratio. Then, we study the properties of the acceptance probability. It is shown that this function is convex for low temperatures and concave for high temperatures. We also provide a lower bound for the number of plateaux of a simulated annealing based on a geometric cooling schedule. Finally, many numerical experiments are reported.