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Rigorous proofs of the convergence of the simulated annealing process in the original formulation have only been given in the case of a finite state space. In Geman and Hwang [5] the idea of a stochastic search of a minimlun was cast in the form of solving a stochastic differential equation describing diffusion, together with a proof of the weak convergence of the process. Kushner [11] and others study generalizations of the process considered in [5]. The aim of this paper is to generalize the definition of the stochastic process corresponding to the original simulated annealing algorithm to the case of an arbitrary state space and study the weak convergence of the process. The results are expressed in terms of the generating distributions and the sequence of successive temperature parameters. At the same time we obtain estimates for the (Dobrushin) coefrcient of ergodicity for an nth iterate of some important kinds of generating distributions. These estimates are also of independent interest. For most of the results the proofs are only sketched here. For full proofs see [6].

Weak convergence of the simulated annealing process in general state space