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Evolutionary Algorithms with Stable Mutations Based on a Discrete Spectral Measure

  • Andrzej Obuchowicz
  • Przemysław Prȩtki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6114)

Abstract

In this paper the concept of multidimensional discrete spectral measure is introduced in the context of its application to real-valued evolutionary algorithms. The notion of discrete spectral measure makes possible to uniquely define a class of multivariate heavy-tailed distributions, that have received more and more attention of evolutionary optimization commynity , recently. Simple sample illustrates advantages of such approach.

Keywords

Evolutionary Algorithm Random Vector Stable Distribution Global Optimization Algorithm Stable Mutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Beyer, H.G., Schwefel, H.P.: Evolutionary strategies – a comprehensive introduction. Neural Computing 1(1), 3–52 (2002)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Byczkowski, T., Nolan, J.P., Rajput, B.: Approximation of multidimensional stable densities. J. of Mult. Anal. 46, 13–31 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury Press (1995)Google Scholar
  4. 4.
    Gutowski, M.: Lévy flights as an underlying mechanism for a global optimization algorithm. In: Proc. 5th Conf. Evolutionary Algorithms and Global Optimization, pp. 79–86. Warsaw University of Technology Press (2001)Google Scholar
  5. 5.
    Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolutionary strategies. Evolutionary Computation 9(2), 159–195 (2001)CrossRefGoogle Scholar
  6. 6.
    Karcz-Dulȩba, I.: Asymptotic behaviour of a discrete dynamical system generated by a simple evolutionary process. Int. Journ. Appl. Math. Comput. Sci. 14(1), 79–90 (2004)MathSciNetGoogle Scholar
  7. 7.
    Kemp, F.: An introduction to sequential Monte Carlo methods. Journal of the Royal Statistical Society D52, 694–695 (2003)Google Scholar
  8. 8.
    Kern, S., Uller, S., Uche, D., Hansen, N., Koumoutsakos, P.: Learning probability distributions in continuous evolutionary algorithms. In: Proc. Workshop on Fundamentals in Evolutionary Algorithms, 13th Int. Colloquium on Automata, Languages and Programming, Eindhoven (2004)Google Scholar
  9. 9.
    Liu, X., Xu, W.: A new filled function applied to global optimization, Comput. Oper. Res. 31, 61–80 (2004)CrossRefGoogle Scholar
  10. 10.
    MacKey, D.C.J.: Introduction to Monte Carlo methods. In: Jordan, M.I. (ed.) Learning in Graphical Models. NATO Science Series, pp. 175–204. Kluwer Academmic Press, Dordrecht (1998)Google Scholar
  11. 11.
    Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, London (1996)zbMATHGoogle Scholar
  12. 12.
    Nolan, J.P., Panorska, A.K., McCulloch, J.H.: Estimation of stable spectral measures - stable non-Gaussian models in finanse and econometrics. Math. Comput. Modelling 34(9), 1113–1122 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Obuchowicz, A.: Evolutionary Algorithms in Global Optimization and Dynamic System Diagnosis. Lubuskie Scientific Society Press, Zielona Góra (2003)Google Scholar
  14. 14.
    Obuchowicz, A., Prȩtki, P.: Phenotypic Evolution with Mutation Based on Symmetric α-Stable Distributions. Int. J. Applied Mathematics and Computer Science 14, 289–316 (2004)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Obuchowicz, A., Prȩtki, P.: Isotropic Symmetric α-Stable Mutations for Evolutionary Algorithms. In: Proc. IEEE Congress on Evoutionary Computation, CEC 2005, pp. 404–410 (2005)Google Scholar
  16. 16.
    Prȩtki, P.: α-Stable Distributions in Evolutionary Algorithms of Parametric Global Optimization. PhD Thesis, University of Zielona Góra, Poland (2008) (in Polish)Google Scholar
  17. 17.
    Rudolph, G.: Local convergence rates of simple evolutionary algorithms wich Cauchy mutations. IEEE Trans. Evolutionary Computation 1(4), 249–258 (1997)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)zbMATHGoogle Scholar
  19. 19.
    Spall, J.C.: Introduction to Stochastic Search and optimization. Wiley, Hoboken (1993)Google Scholar
  20. 20.
    Vidysagar, M.: Randomized algorithms for robust controller synthesis using statistical learning theory. Automatica 37, 1515–1528 (2001)CrossRefGoogle Scholar
  21. 21.
    Yao, X., Liu, Y.: Fast evolution strategies. In: Angeline, P.J., McDonnell, J.R., Reynolds, R.G., Eberhart, R. (eds.) EP 1997. LNCS, vol. 1213, pp. 151–161. Springer, Heidelberg (1997)Google Scholar
  22. 22.
    Yao, X., Liu, Y., Liu, G.: Evolutionary Programming made faster. IEEE Trans. Evolutionary Computation 3(2), 82–102 (1999)CrossRefGoogle Scholar
  23. 23.
    Zolotariev, A.: One-Dimensional Stable Distributions. American Mathematical Society, Providence (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Andrzej Obuchowicz
    • 1
  • Przemysław Prȩtki
    • 1
  1. 1.Institute of Control and Computation EngineeringUniversity of Zielona GóraZielona GóraPoland

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