The Annals of Applied Probability

Piecewise constant triangular cooling schedules for generalized simulated annealing algorithms

Olivier Catoni and Cécile Cot

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Abstract

We investigate how to tune a generalized simulated annealing algorithm with piecewise constant cooling schedule to get an optical convergence exponent. The optimal convergence exponent of generalized simulated annealing algorithms has been computed by Catoni and Trouvé. It is reached only with triangular sequences of temperatures, meaning that different finite sequences are used, depending on the time resource available for computations (expressed by an overall number of iterations). We show first that it is possible to get close to the optimal convergence exponent uniformly over suitably bounded families of energy landscapes using a fixed number of temperature steps. Then we show that, letting the number of steps increase with the time resource, we can build a cooling schedule which is universally robust with respect to the convergence exponent: a fixed triangular sequence of temperatures gives an optimal convergence exponent for any energy landscape. Piecewise constant temperature sequences are often used in practice: in favourable cases, the use of the same temperature during a large number of iterations allows tabulating the exponential penalties appearing in the transition matrix, thus sparing a significant amount of computer time. The proofs we give rely on Freidlin and Wentzell's closed formulas for the exit time and point from subdomains of time homogeneous Markov chains.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 2 (1998), 375-396.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903532

Digital Object Identifier
doi:10.1214/aoap/1028903532

Mathematical Reviews number (MathSciNet)
MR1624937

Zentralblatt MATH identifier
1053.65500

Subjects
Primary: 82C80: Numerical methods (Monte Carlo, series resummation, etc.) 65C05: Monte Carlo methods
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90C42 60F10: Large deviations

Keywords
Simulated annealing Metropolis dynamic Markov chains with rare transitions triangular cooling schedules optimal convergence exponent

Citation

Cot, Cécile; Catoni, Olivier. Piecewise constant triangular cooling schedules for generalized simulated annealing algorithms. Ann. Appl. Probab. 8 (1998), no. 2, 375--396. doi:10.1214/aoap/1028903532. https://projecteuclid.org/euclid.aoap/1028903532


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  • DMI, LABORATOIRE DE MATHEMATIQUES DIAM, LABORATOIRE DE MATHEMATIQUES ´ ´ DE L 'ECOLE NORMALE SUPERIEURE DE L 'ECOLE NORMALE SUPERIEURE ´ ´ UA 762 DU CNRS UA 762 DU CNRS 45 RUE D'ULM 45 RUE D'ULM 75230 PARIS CEDEX 05 75230 PARIS CEDEX 05 FRANCE FRANCE AND E-MAIL: catoni@dmi.ens.fr. LABORATOIRE DE MODELISATION STOCHASTIQUE ´ ET STATISTIQUE UNIVERSITE PARIS SUD ´ MATHEMATIQUES ´ BATIMENT 425 91405 ORSAY CEDEX FRANCE E-MAIL: cot@dmi.ens.fr cot@stats.matups.fr