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A review of distances for the Mallows and Generalized Mallows estimation of distribution algorithms

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Abstract

The Mallows (MM) and the Generalized Mallows (GMM) probability models have demonstrated their validity in the framework of Estimation of distribution algorithms (EDAs) for solving permutation-based combinatorial optimisation problems. Recent works, however, have suggested that the performance of these algorithms strongly relies on the distance used under the model. The goal of this paper is to review three common distances for permutations, Kendall’s-\(\tau \), Cayley and Ulam, and compare their performance under MM and GMM EDAs. Moreover, with the aim of predicting the most suitable distance for solving any given permutation problem, we focus our attention on the relation between these distances and the neighbourhood systems in the field of local search optimisation. In this sense, we demonstrate that the performance of the MM and GMM EDAs is strongly correlated with that of multistart local search algorithms when using related neighbourhoods. Furthermore, by means of fitness landscape analysis techniques, we show that the suitability of a distance to solve a problem is clearly characterised by the generation of high smoothness fitness landscapes.

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Notes

  1. With parameter we refer to the smallest component of an instance that is used to calculate the fitness of a solution.

  2. Supplementary results, source codes, instances, and extended material of the experiments can be downloaded from http://www.sc.ehu.es/ccwbayes/members/jceberio/COAP/Review_Distances.html.

  3. The ARPD results have been calculated using the best known results obtained by the MLSs. Results from the previous section have not been used here.

  4. For the specific expressions we refer the interested reader to address the original paper [37].

References

  1. Baker, K.: Introduction to Sequencing and Scheduling. Wiley, Hoboken (1974)

    Google Scholar 

  2. Box, G.E.P., Jenkins, G.: Time Series Analysis, Forecasting and Control. Holden-Day, Incorporated, San Francisco (1970)

    MATH  Google Scholar 

  3. Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: A review on estimation of distribution algorithms in permutation-based combinatorial optimization problems. Prog. Artif. Intell. 1(1), 103–117 (2012)

    Article  Google Scholar 

  4. Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: A distance-based ranking model estimation of distribution algorithm for the flowshop scheduling problem. IEEE Trans. Evol. Comput. 18(2), 286–300 (2014)

    Article  Google Scholar 

  5. Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: Extending distance-based ranking models in estimation of distribution algorithms. In: Proceedings of the 2014 IEEE Congress on Evolutionary Computation. IEEE (2014)

  6. Ceberio, J., Mendiburu, A., Lozano, J.A.: Introducing the Mallows Model on Estimation of Distribution Algorithms. In: Lu B.L., Zhang L., Kwok J.T. (eds.) Proceedings of International Conference on Neural Information Processing (ICONIP). Lecture Notes in Computer Science, pp. 461–470. Springer (2011)

  7. Ceberio, J., Mendiburu, A., Lozano, J.A.: The Plackett-Luce ranking model on permutation-based optimization problems. In: Proceedings of the 2013 IEEE Congress on Evolutionary Computation, pp. 494–501 (2013)

  8. Ceberio, J., Mendiburu, A., Lozano, J.A.: The linear ordering problem revisited. Eur. J. Oper. Res. 241(3), 686–696 (2014)

    Article  MathSciNet  Google Scholar 

  9. Diaconis, P.: Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward (1988)

    MATH  Google Scholar 

  10. Fligner, M., Verducci, J.: Multistage ranking models. J. Am. Stat. Assoc. 83(403), 892–901 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Garcia, S., Herrera, F.: An extension on “statistical comparisons of classifiers over multiple data set” for all pairwise comparisons. J. Mach. Learn. Res. 9, 2677–2694 (2008)

    MATH  Google Scholar 

  12. Garcia, S., Molina, D., Lozano, M., Herrera, F.: A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 Special Session on Real Parameter Optimization. J. Heuristics 15(6), 617–644 (2009)

    Article  MATH  Google Scholar 

  13. Goldberg, D.E., Jr., Lingle, R.: Alleles Loci and the Traveling Salesman Problem. In: Proceedings of an International Conference on Genetic Algorithms and Their Applications, vol. 154, pp. 154–159. Lawrence Erlbaum, Hillsdale, NJ (1985)

  14. Inza, I., Larrañaga, P., Sierra, B.: Feature subset selection by estimation of distribution algorithms. In: Larrañaga, P., Lozano, J.A. (eds.) Estimation of Distribution Algorithms. A New Tool for Evolutionary Computation, pp. 269–294. Kluwer Academic Publishers, Dordrecht (2002)

    Chapter  Google Scholar 

  15. Irurozki, E., Calvo, B., Lozano, J.A.: An r package for permutations, Mallows and Generalized Mallows models. Technical Report, University of the Basque Country UPV/EHU (2014)

  16. Irurozki, E., Calvo, B., Lozano, J.A.: Sampling and learning Mallows and Generalized Mallows models under the Cayley distance. Technical Report, University of the Basque Country UPV/EHU (2014)

  17. Irurozki, E., Calvo, B., Lozano, J.A.: Sampling and learning the Mallows model under the ulam distance. Technical Report, University of the Basque Country UPV/EHU (2014)

  18. Koopmans, T.C., Beckmann, M.J.: Assignment Problems and the Location of Economic Activities. Cowles Foundation Discussion Papers 4, Cowles Foundation for Research in Economics, Yale University (1955)

  19. Larrañaga, P., Lozano, J.A.: Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Dordrecht (2002)

    Book  Google Scholar 

  20. Lozano, J.A., Larrañaga, P., Inza, I., Bengoetxea, E.: Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms (Studies in Fuzziness and Soft Computing). Springer, Secaucus (2006)

    Book  Google Scholar 

  21. Lozano, J.A., Mendiburu, A.: Estimation of distribution algorithms applied to the job schedulling problem. In: Larrañaga, P., Lozano, J.A. (eds.) Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  22. Manderick, B., de Weger, M., Spiessens, P.: The genetic algorithm and the structure of the fitness landscape. In: Proceedings of the 4th International Conference on Genetic Algorithms, pp. 143–150 (1991)

  23. Mandhani, B., Meila, M.: Tractable search for learning exponential models of rankings. J. Mach. Learn. Res. 5, 392–399 (2009)

    Google Scholar 

  24. Marti, R.: Multi-start methods. In: Glover, F., Kochenberger, G. (eds.) Handbook of Metaheuristics, International Series in Operations Research and Management Science, vol. 57, pp. 355–368. Springer, Berlin (2003)

    Chapter  Google Scholar 

  25. Martí, R., Reinelt, G.: The Linear Ordering Problem: Exact and Heuristic Methods in Combinatorial Optimization, vol. 175. Springer, Berlin (2011)

    Google Scholar 

  26. Pelikan, M., Goldberg, D.E., Lobo, F.G.: A survey of optimization by building and using probabilistic models. Comput. Optim. Appl. 21(1), 5–20 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Reinelt, G.: TSPLIB—A T.S.P. library. Technical Report 250, Universität Augsburg, Institut für Mathematik, Augsburg (1990)

  28. Robles, V., de Miguel, P., Larrañaga, P.: Solving the traveling salesman problem with EDAs. In: Larrañaga, P., Lozano, J.A. (eds.) Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  29. Santana, R., Bielza, C., Larranaga, P., Lozano, J.A., Echegoyen, C., Mendiburu, A., Armananzas, R., Shakya, S.: Mateda-2.0: estimation of distribution algorithms in matlab. J. Stat. Softw. 35(7), 1–30 (2010)

    Article  Google Scholar 

  30. Santana, R., Larrañaga, P., Lozano, J.A.: Combining variable neighborhood search and estimation of distribution algorithms in the protein side chain placement problem. J. Heuristics 14(5), 519–547 (2008)

    Article  MATH  Google Scholar 

  31. Santana, R., Larrañaga, P., Lozano, J.A.: Protein folding in simplified models with estimation of distribution algorithms. IEEE Trans. Evol. Comput. 12(4), 418–438 (2008)

    Article  Google Scholar 

  32. Schiavinotto, T., Stützle, T.: The linear ordering problem: instances, search space analysis and algorithms. J. Math. Model. Algorithms 3, 367–402 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Schiavinotto, T., Stützle, T.: A review of metrics on permutations for search landscape analysis. Comput. Oper. Res. 34(10), 3143–3153 (2007)

    Article  MATH  Google Scholar 

  34. Taillard, E.: Benchmarks for basic scheduling problems. Eur. J. Oper. Res. 64(2), 278–285 (1993)

    Article  MATH  Google Scholar 

  35. Taillard, E.: Problem instances (1993). http://mistic.heig-vd.ch/taillard/problemes.dir/problemes.html

  36. Tsutsui, S., Wilson, G.: Solving Capacitated Vehicle Routing Problems Using Edge Histogram Based Sampling Algorithms. In: Proceedings of the IEEE Conference on Evolutionary Computation, pp. 1150–1157. Portland, Oregon (2004)

  37. Vassilev, V.K., Fogarty, T.C., Miller, J.F.: Information characteristics and the structure of landscapes. Evol. Comput. 8(1), 31–60 (2000)

    Article  Google Scholar 

  38. Weinberger, E.: Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern. 63(5), 325–336 (1990)

    Article  MATH  Google Scholar 

  39. Zhang, Q., Sun, Q., Tsang, E.: An evolutionary algorithm with guided mutation for the maximum clique problem. IEEE Trans. Evol. Comput. 9, 192–200 (2005)

    Article  Google Scholar 

  40. Zhang, Q., Sun, J., Tsang, E., Ford, J.: Estimation of distribution algorithm with 2-opt local search for the quadratic assignment problem. Stud. Fuzziness Soft Comput. 192(2006), 281–292 (2006)

    Article  Google Scholar 

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Acknowledgments

This work has been partially supported by the Saiotek and Research Groups 2013-2018 (IT-609-13) programs (Basque Government), TIN2010-14931 (Ministry of Science and Technology), COMBIOMED network in computational bio-medicine (Carlos III Health Institute) and the NICaiA Project PIRSES-GA-2009-247619 (European Commission). Josu Ceberio holds a grant from the Basque Government.

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Authors and Affiliations

  1. Intelligent Systems Group, Department of Computer Science and Artificial Intelligence, University of the Basque Country UPV/EHU, Donostia, Spain

    Josu Ceberio, Ekhine Irurozki & Jose A. Lozano

  2. Intelligent Systems Group, Department of Computer Architecture and Technology, University of the Basque Country UPV/EHU, Donostia, Spain

    Alexander Mendiburu

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  1. Josu Ceberio
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  2. Ekhine Irurozki
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  3. Alexander Mendiburu
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  4. Jose A. Lozano
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Correspondence to Josu Ceberio.

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Ceberio, J., Irurozki, E., Mendiburu, A. et al. A review of distances for the Mallows and Generalized Mallows estimation of distribution algorithms. Comput Optim Appl 62, 545–564 (2015). https://doi.org/10.1007/s10589-015-9740-x

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