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

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  • R. Y. Rubinstein

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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.

Suggested Citation

  • R. Y. Rubinstein, 2005. "A Stochastic Minimum Cross-Entropy Method for Combinatorial Optimization and Rare-event Estimation," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 5-50, March.
    • Handle: RePEc:spr:metcap:v:7:y:2005:i:1:d:10.1007_s11009-005-6653-7
    DOI: 10.1007/s11009-005-6653-7
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    References listed on IDEAS

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    1. Reuven Rubinstein, 1999. "The Cross-Entropy Method for Combinatorial and Continuous Optimization," Methodology and Computing in Applied Probability, Springer, vol. 1(2), pages 127-190, September.
    2. Pieter-Tjerk de Boer & Dirk Kroese & Shie Mannor & Reuven Rubinstein, 2005. "A Tutorial on the Cross-Entropy Method," Annals of Operations Research, Springer, vol. 134(1), pages 19-67, February.
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    Cited by:

    1. Zdravko I. Botev & Dirk P. Kroese, 2008. "An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 471-505, December.
    2. Reuven Y. Rubinstein, 2006. "How Many Needles are in a Haystack, or How to Solve #P-Complete Counting Problems Fast," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 5-51, March.
    3. Ad Ridder & Thomas Taimre, 2011. "State-dependent importance sampling schemes via minimum cross-entropy," Annals of Operations Research, Springer, vol. 189(1), pages 357-388, September.
    4. Zdravko I. Botev & Dirk P. Kroese, 2008. "Non-asymptotic Bandwidth Selection for Density Estimation of Discrete Data," Methodology and Computing in Applied Probability, Springer, vol. 10(3), pages 435-451, September.
    5. Zdravko I. Botev & Dirk P. Kroese, 2011. "The Generalized Cross Entropy Method, with Applications to Probability Density Estimation," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 1-27, March.
    6. Reuven Rubinstein, 2008. "Semi-Iterative Minimum Cross-Entropy Algorithms for Rare-Events, Counting, Combinatorial and Integer Programming," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 121-178, June.

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