skip to main content
research-article

Particle Algorithms for Optimization on Binary Spaces

Published: 01 January 2013 Publication History

Abstract

We discuss a unified approach to stochastic optimization of pseudo-Boolean objective functions based on particle methods, including the cross-entropy method and simulated annealing as special cases. We point out the need for auxiliary sampling distributions, meaning parametric families on binary spaces, which are able to reproduce complex dependency structures, and illustrate their usefulness in our numerical experiments. We provide numerical evidence that particle-driven optimization algorithms based on parametric families yield superior results on strongly multimodal optimization problems while local search heuristics outperform them on easier problems.

Supplementary Material

PDF File (a8-schafer_appendix.pdf)
The proof is given in an electronic appendix, available online in the ACM Digital Library.

References

[1]
Alidaee, B., Kochenberger, G., and Wang, H. 2010. Theorems supporting r-flip search for pseudo-Boolean optimization. Int. J. Appl. Metaheur. Comput. (IJAMC) 1, 1, 93--109.
[2]
Amini, M., Alidaee, B., and Kochenberger, G. 1999. A scatter search approach to unconstrained quadratic binary programs. In New Ideas in Optimization. McGraw-Hill Ltd., UK, 317--330.
[3]
Beasley, J. 1990. OR-Library: Distributing test problems by electronic mail. J. Operat. Res. Soc., 1069--1072.
[4]
Beasley, J. 1998. Heuristic algorithms for the unconstrained binary quadratic programming problem. Tech. rep., Management School, Imperial College London.
[5]
Billionnet, A. and Sutter, A. 1994. Minimization of a quadratic pseudo-Boolean function. Europ. J. Oper. Res. 78, 1, 106--115.
[6]
Boros, E. and Hammer, P. 2002. Pseudo-Boolean optimization. Discr. Appl. Math. 123, 1--3, 155--225.
[7]
Boros, E., Hammer, P., and Tavares, G. 2007. Local search heuristics for quadratic unconstrained binary optimization (QUBO). J. Heuris. 13, 2, 99--132.
[8]
Carpenter, J., Clifford, P., and Fearnhead, P. 1999. Improved Particle Filter for nonlinear problems. IEE Proc. Radar, Sonar Nav. 146, 1, 2--7.
[9]
Chopin, N. 2002. A sequential particle filter method for static models. Biometrika 89, 3, 539.
[10]
Del Moral, P., Doucet, A., and Jasra, A. 2006. Sequential Monte Carlo samplers. J. Roy. Stat. Soc. Ser. B (Stat. Meth.) 68, 3, 411--436.
[11]
Del Moral, P., Doucet, A., and Jasra, A. 2012. An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. to appear.
[12]
Emrich, L. and Piedmonte, M. 1991. A method for generating high-dimensional multivariate binary variates. Amer. Statist. 45, 302--304.
[13]
Fearnhead, P. and Clifford, P. 2003. Online inference for hidden Markov models via particle filters. J. Roy. Stat. Soc. Ser. B (Stat. Meth.) 65, 4, 887--899.
[14]
Garey, M. and Johnson, D. 1979. Computers and Intractability: A Guide to the Theory of NP-completeness. WH Freeman & Co.
[15]
Genz, A. and Bretz, F. 2009. Computation of Multivariate Normal and T Probabilities. Vol. 195. Springer.
[16]
Glover, F., Kochenberger, G., and Alidaee, B. 1998. Adaptive memory tabu search for binary quadratic programs. Manage. Sci. 44, 336--345.
[17]
Gordon, N. J., Salmond, D. J., and Smith, A. F. M. 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. Radar, Sonar Navigation 140, 2, 107--113.
[18]
Gueye, S. and Michelon, P. 2009. A linearization framework for unconstrained quadratic (0-1) problems. Discr. Appl. Math. 157, 6, 1255--1266.
[19]
Hansen, P. and Meyer, C. 2009. Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem. Discr. Appl. Math. 157, 6, 1267--1290.
[20]
Hasan, M., AlKhamis, T., and Ali, J. 2000. A comparison between simulated annealing, genetic algorithm and tabu search methods for the unconstrained quadratic Pseudo-Boolean function. Comput. Indust. Eng. 38, 3, 323--340.
[21]
Jasra, A., Stephens, D., Doucet, A., and Tsagaris, T. 2011. Inference for Lévy-Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo. Scand. J. Stat.
[22]
Joe, H. 1996. Families of m-variate distributions with given margins and m (m-1)/2 bivariate dependence parameters. Lecture Notes-Monograph Series 28, 120--141.
[23]
Johansen, A., Moral, P. D., and Doucet, A. 2006. Sequential Monte Carlo samplers for rare events. In Proceedings of the 6th International Workshop on Rare Event Simulation. 256--267.
[24]
Katayama, K. and Narihisa, H. 2001. Performance of Simulated Annealing-based heuristic for the unconstrained binary quadratic programming problem. Europ. J. Operat. Res. 134, 1, 103--119.
[25]
Kirkpatrick, S., Gelatt, C., and Vecchi, M. 1983. Optimization by simulated annealing. Science 220, 4598, 671.
[26]
Kitagawa, G. 1996. Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Computat. Graph. Stat. 5, 1, 1--25.
[27]
Kong, A., Liu, J. S., and Wong, W. H. 1994. Sequential imputation and Bayesian missing data problems. JASA 89, 278--288.
[28]
Kozlov, M., Tarasov, S., and Khachiyan, L. 1979. Polynomial solvability of convex quadratic programming. In Soviet Mathematics Doklady 20, 1108--1111.
[29]
Liu, J. and Chen, R. 1998. Sequential Monte Carlo methods for dynamic systems. JASA 93, 443, 1032--1044.
[30]
Merz, P. and Freisleben, B. 1999. Genetic algorithms for binary quadratic programming. In Proceedings of the Genetic and Evolutionary Computation Conference. Vol. 1. Citeseer, 417--424.
[31]
Merz, P. and Freisleben, B. 2002. Greedy and local search heuristics for unconstrained binary quadratic programming. J. Heur. 8, 2, 197--213.
[32]
Merz, P. and Katayama, K. 2004. Memetic algorithms for the unconstrained binary quadratic programming problem. BioSystems 78, 1--3, 99--118.
[33]
Nott, D. and Kohn, R. 2005. Adaptive sampling for Bayesian variable selection. Biometrika 92, 4, 747.
[34]
Palubeckis, G. 1995. A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming. Computing 54, 4, 283--301.
[35]
Palubeckis, G. 2004. Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann. Oper. Res. 131, 1, 259--282.
[36]
Pardalos, P. 1991. Construction of test problems in quadratic bivalent programming. ACM Trans. Math. Softw. (TOMS) 17, 1, 74--87.
[37]
Pardalos, P. and Rodgers, G. 1990. Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 2, 131--144.
[38]
Poljak, S. and Wolkowicz, H. 1995. Convex relaxations of (0, 1)-quadratic programming. Math. Oper. Res., 550--561.
[39]
Robert, C. and Casella, G. 2004. Monte Carlo Statistical Methods. Springer Verlag.
[40]
Rubinstein, R. Y. 1997. Optimization of computer simulation models with rare events. Europ. J. Oper. Res. 99, 89--112.
[41]
Rubinstein, R. Y. 1999. The Cross-Entropy Method for combinatorial and continuous optimization. Methodology and Computing in Applied Probability 1, 127--190.
[42]
Rubinstein, R. Y. and Kroese, D. P. 2004. The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning. Springer-Verlag.
[43]
Schäfer, C. 2010. Parametric families on binary spaces. Tech. rep. arXiv:1008.0055.
[44]
Schäfer, C. 2012. On parametric families for sampling binary data with specified mean and correlation. Tech. rep. arXiv:1111.0576.
[45]
Schäfer, C. and Chopin, N. 2012. Sequential Monte Carlo on large binary sampling spaces. Stat. Comput. to appear.

Cited By

View all

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Modeling and Computer Simulation
ACM Transactions on Modeling and Computer Simulation  Volume 23, Issue 1
Special Issue on Monte Carlo Methods in Statistics
January 2013
207 pages
ISSN:1049-3301
EISSN:1558-1195
DOI:10.1145/2414416
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2013
Accepted: 01 September 2012
Revised: 01 July 2012
Received: 01 October 2011
Published in TOMACS Volume 23, Issue 1

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. Binary parametric families
  2. cross-entropy method
  3. pseudo-Boolean optimization
  4. sequential Monte Carlo
  5. simulated annealing

Qualifiers

  • Research-article
  • Research
  • Refereed

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)9
  • Downloads (Last 6 weeks)1
Reflects downloads up to 14 Sep 2024

Other Metrics

Citations

Cited By

View all

View Options

Get Access

Login options

Full Access

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media