Convergence theorems for a class of simulated annealing algorithms on ℝ

The paper studies a class of simulated annealing algorithms for global minimization of a continuous function defined on a subset of ℝ^d. It considers the case where the selection Markov kernel is absolutely continuous and has a density which is uniformly bounded away from 0. This class includes certain simulated annealing algorithms recently introduced by various authors. The paper shows that, under mild conditions, the sequence of states generated by these algorithms converges in probability to the global minimum of the function. Unlike most previous studies where the cooling schedule is deterministic, the present cooling schedule is allowed to be adaptive. The paper also addresses the issue of almost sure convergence versus convergence in probability.