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


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.

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  • E. H. L. Aarts and J. H. M. Korst, Simulated Annealing and Boltzmann Machines, John Wiley & Sons, 1989.

  • T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, Inc, 1991.

  • 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).

  • D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison Wesley, 1989.

  • J. N. Kapur, and H. K. Kesavan, Entropy Optimization with, Applications, Academic Press, Inc., 1992.

  • J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer: Berlin, Heidelberg, New York, 2001.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • R. Y. Rubinstein and B. Melamed, Modern Simulation and Modeling, John Wiley & Sons, Inc., 1998.

  • H. D. Wolpert, Information Theory—The Bridge Connecting Bounded Rational Game Theory and Statistical Physics. Manuscript, NASA Ames Research Center, in press.

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Correspondence to R. Y. Rubinstein.

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AMS 2000 Subject Classification: 65C05, 60C05, 68W20, 90C59

*This reseach was supported by the Israel Science Foundation (grant no 191-565).

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Rubinstein, R.Y. A Stochastic Minimum Cross-Entropy Method for Combinatorial Optimization and Rare-event Estimation*. Methodol Comput Appl Probab 7, 5–50 (2005).

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