Abstract
It is well known that the behaviour of the simulated annealing approach to optimization is crucially dependent on the choice of temperature schedule. In this paper, a dynamic programming approach is used to find the temperature schedule which is optimal for a simple minimization problem. The optimal schedule is compared with certain standard non-optimal choices. These generally perform well provided the first and last temperatures are suitably selected. Indeed, these temperatures can be chosen in such a way as to make the performance of the logarithmic schedule almost optimal. This optimal performance is fairly robust to the choice of the first temperature.
The dynamic programming approach cannot be applied directly to problems of more realistic size, such as those arising in statistical image reconstruction. Nevertheless, some simulation experiments suggest that the general conclusions from the simple minimization problem do carry over to larger problems. Various families of schedules can be made to perform well with suitable choice of the first and last temperatures, and the logarithmic schedule combines good performance with reasonable robustness to the choice of the first temperature.
Similar content being viewed by others
References
Bellman, R. (1957) Dynamic Programming, Princeton University Press, Princeton, NJ.
Bertsimas, T. and Tsitsiklis, J. (1993) Simulated annealing. Statistical Science, 8, 10–15.
Besag, J. E. (1986) On the statistical analysis of dirty pictures (with discussion). Journal of the Royal Statistical Society B, 48, 259–302.
Geman, D. and Geman, S. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
Geman, S. and McClure, D. E. (1987) Statistical methods for tomographic image reconstruction. Bulletin of the International Statistical Institute, 52 (4), 5–21.
Geman, D. and Reynolds, G. (1992) Constrained restoration and the recovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 164–180.
Greig, D.M., Porteous, B.T. and Seheult, A.H. (1989) Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society B, 51, 271–279.
Hajek, B. (1988) Cooling schedules for optimal annealing. Mathematics of Operational Research, 13, 311–329.
Laarhoven, P. J. M. and Aarts, E. H. L. (1987) Simulated Annealing: Theory and Applications, D. Reidel, Dordrecht.
Ripley, B. D. (1987) Stochastic Simulation, Wiley, New York.
Ripley, B. D. (1988) Statistical Inference for Spatial Processes, Cambridge University Press, Cambridge.
Sibson, R. (1987) CONICON3 Handbook, School of Mathematical Sciences, University of Bath.
Stander, J. and Silverman, B. W. (1992) Temperature schedules for simulated annealing with particular reference to image reconstruction. Quaderno n. 16/1992, IAC, CNR, Rome.
Wichmann, B. A. and Hill, J. D. (1982) Algorithm AS183. An efficient and portable pseudo-random number generator. Applied Statistics, 31, 188–190.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stander, J., Silverman, B.W. Temperature schedules for simulated annealing. Stat Comput 4, 21–32 (1994). https://doi.org/10.1007/BF00143921
Issue Date:
DOI: https://doi.org/10.1007/BF00143921