On logarithmic simulated annealing

Abstract

We perform a convergence analysis of simulated annealing for the special case of logarithmic cooling schedules. For this class of simulated annealing algorithms, B. Hajek [7] proved that the convergence to optimum solutions requires the lower bound F = ln (k + 2) on the cooling schedule, where k is the number of transitions and ε denotes the maximum value of the escape depth from local minima. Let n be a uniform upper bound for the number of neighbours in the underlying configuration space. Under some natural assumptions, we prove the following convergence rate: After k nO(ε) + logO(1) (1=") transitions the probability to be in an optimum solution is larger than (1-"). The result can be applied, for example, to the average case analysis of stochastic algorithms when estimations of the corresponding values ε are known. © Springer-Verlag Berlin Heidelberg 2002.