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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 proved that the convergence to optimum solutions requires the lower bound Γ/ln (k + 2) on the cooling schedule, where k is the number of transitions and Γ 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(Γ) + logO(1) (1/Ɛ) transitions the probability to be in an optimum solution is larger than (1 - Ɛ). The result can be applied, for example, to the average case analysis of stochastic algorithms when estimations of the corresponding values Γ are known.