Advertisement

A Stochastic Minimum Cross-Entropy Method for Combinatorial Optimization and Rare-event Estimation*

  • R. Y. Rubinstein
Original Article

Abstract

We present a new method, called the minimum cross-entropy (MCE) method for approximating the optimal solution of NP-hard combinatorial optimization problems and rare-event probability estimation, which can be viewed as an alternative to the standard cross entropy (CE) method. The MCE method presents a generic adaptive stochastic version of Kull-back’s classic MinxEnt method. We discuss its similarities and differences with the standard cross-entropy (CE) method and prove its convergence. We show numerically that MCE is a little more accurate than CE, but at the same time a little slower than CE. We also present a new method for trajectory generation for TSP and some related problems. We finally give some numerical results using MCE for rare-events probability estimation for simple static models, the maximal cut problem and the TSP, and point out some new areas of possible applications.

Keywords

combinatorial optimization cross-entropy rare-event estimation Monte Carlo simulation stochastic optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. H. L. Aarts and J. H. M. Korst, Simulated Annealing and Boltzmann Machines, John Wiley & Sons, 1989.Google Scholar
  2. T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc, 1991.Google Scholar
  3. P. T. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, A Tutorial on the Cross-Entropy Method, Annals of Operations Research, 2005, (to appear).Google Scholar
  4. D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, 1989.Google Scholar
  5. J. N. Kapur, and H. K. Kesavan, Entropy Optimization with, Applications, Academic Press, Inc., 1992.Google Scholar
  6. J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer: Berlin, Heidelberg, New York, 2001.Google Scholar
  7. R. Y. Rubinstein, “The cross-entropy method for combinatorial and continuous optimization,” Methodology and Computing in Applied Probability vol. 2, pp. 127–190, 1999.Google Scholar
  8. R. Y. Rubinstein, “Cross-entropy and rare event formula-native maximal cul and bipartition problems,” ACM Transactions on Modelling and Computer Simulation vol. 12(1) pp. 27–53, 2002.Google Scholar
  9. R. Y. Rubinstein, and D. P. Kroese, The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning, Springer: Berlin, Heidelberg, New York, 2004.Google Scholar
  10. R. Y. Rubinstein and B. Melamed, Modern Simulation and Modeling, John Wiley & Sons, Inc., 1998.Google Scholar
  11. H. D. Wolpert, Information Theory—The Bridge Connecting Bounded Rational Game Theory and Statistical Physics. Manuscript, NASA Ames Research Center, in press.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Faculty of Industrial Engineering and ManagementTechnionHaifaIsrael

Personalised recommendations