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Applications of sharp large deviations estimates to optimal cooling schedules. (English) Zbl 0752.60025
This paper is a sequel to [author, ibid. 27, No. 3, 291-383 (1991; Zbl 0746.60024)]. It has been drawn from the author’s thesis at the University Paris-Sud Orsay, March 1990, supervised by R. Azencott. Aim of this second part is to study applications of large deviations to cooling systems of the critical type, i.e. $1/{T}_{n}=\left(1/d\right)lnn+B+o\left(1\right)$, where $d$ is Hajek’s critical depth. Although quasi-equilibrium is not maintained for such schedules, it turns out that the law of the system is not “too far” from quasi-equilibrium if $B$ is small. However, if $B$ is above some critical value, convergence rates of the annealing algorithm can be made arbitrarily poor by increasing $B$. Sharp large deviations estimates are needed in order to obtain the desired results. - – Contents: 1. Estimation of the probability of the critical cycle. 2. Asymptotics of the law of the system. 3. Triangular cooling schedules. 4. The optimization problem far from the horizon.

##### MSC:
 60F10 Large deviations 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 93E25 Computational methods in stochastic control