Optimal Weighted Recombination

  • Dirk V. Arnold
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3469)


Weighted recombination is a means for improving the local search performance of evolution strategies. It aims to make effective use of the information available, without significantly increasing computational costs per time step. In this paper, the potential speed-up resulting from using rank-based weighted recombination is investigated. Optimal weights are computed for the sphere model, and comparisons with the performance of strategies that do not make use of weighted recombination are presented. It is seen that unlike strategies that rely on unweighted recombination and truncation selection, weighted multirecombination evolution strategies are able to improve on the serial efficiency of the (1+1)-ES on the sphere. The implications of the use of weighted recombination for noisy optimization are studied, and parallels to the use of rescaled mutations are drawn. The cumulative step length adaptation mechanism is formulated for the case of an optimally weighted evolution strategy, and its performance is analyzed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, D.V.: Noisy Optimization with Evolution Strategies. Genetic Algorithms and Evolutionary Computation Series. Kluwer Academic Publishers, Boston (2002)MATHGoogle Scholar
  2. 2.
    Arnold, D.V.: An analysis of evolutionary gradient search. In: Proc. of the 2004 IEEE Congress on Evolutionary Computation, pp. 47–54. IEEE Press, Piscataway (2004)CrossRefGoogle Scholar
  3. 3.
    Arnold, D.V., Beyer, H.-G.: Local performance of the (μ/μI, λ)-ES in a noisy environment. In: Martin, W.N., Spears, W.M. (eds.) Foundations of Genetic Algorithms, vol. 6, pp. 127–141. Morgan Kaufmann Publishers, San Francisco (2001)CrossRefGoogle Scholar
  4. 4.
    Arnold, D.V., Beyer, H.-G.: Performance analysis of evolution strategies with multi-recombination in high-dimensional IRN-search spaces disturbed by noise. Theoretical Computer Science 289(1), 629–647 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arnold, D.V., Beyer, H.-G.: A comparison of evolution strategies with other direct search methods in the presence of noise. Computational Optimization and Applications 24(1), 135–159 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Arnold, D.V., Beyer, H.-G.: Performance analysis of evolutionary optimization with cumulative step length adaptation. IEEE Transactions on Automatic Control 49(4), 617–622 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Balakrishnan, N., Rao, C.R.: Order statistics: An introduction. In: Balakrishnan, N., Rao, C.R. (eds.) Handbook of Statistics, vol. 16, pp. 3–24. Elsevier, Amsterdam (1998)Google Scholar
  8. 8.
    Beyer, H.-G.: Toward a theory of evolution strategies: On the benefit of sex – the (μ/ μ, λ)-theory. Evolutionary Computation 3(1), 81–111 (1995)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Beyer, H.-G.: On the asymptotic behavior of multirecombinant evolution strategies. In: Voigt, H.-M., Ebeling, W., Rechenberg, I., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature – PPSN IV, pp. 122–133. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    Beyer, H.-G.: Mutate Large, But Inherit Small! On the Analysis of Rescaled Mutations in (1̃, λ̃)-ES with Noisy Fitness Data. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 109–118. Springer, Heidelberg (1998)Google Scholar
  11. 11.
    Beyer, H.-G.: Evolutionary algorithms in noisy environments: Theoretical issues and guidelines for practice. Computer Methods in Mechanics and Applied Engineering 186, 239–267 (2000)MATHCrossRefGoogle Scholar
  12. 12.
    Beyer, H.-G.: The Theory of Evolution Strategies. Natural Computing Series. Springer, Heidelberg (2001)Google Scholar
  13. 13.
    David, H.A., Nagaraja, H.N.: Concomitants of order statistics. In: Balakrishnan, N., Rao, C.R. (eds.) Handbook of Statistics, vol. 16, pp. 487–513. Elsevier, Amsterdam (1998)Google Scholar
  14. 14.
    Hansen, N.: Verallgemeinerte individuelle Schrittweitenregelung in der Evolutionsstrategie. Mensch & Buch Verlag, Berlin (1998)Google Scholar
  15. 15.
    Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation 9(2), 159–195 (2001)CrossRefGoogle Scholar
  16. 16.
    Kern, S., Müller, S.D., Hansen, N., Büche, D., Ocenasek, J., Koumoutsakos, P.: Learning probability distributions in continuous evolutionary algorithms – A comparative review. Natural Computing 3(1), 77–112 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ostermeier, A., Gawelczyk, A., Hansen, N.: Step-size adaptation based on nonlocal use of selection information. In: Davidor, Y., Schwefel, H.-P., Männer, R. (eds.) Parallel Problem Solving from Nature – PPSN III, pp. 189–198. Springer, Heidelberg (1994)Google Scholar
  18. 18.
    Rechenberg, I.: Evolutionsstrategie – Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Friedrich Frommann Verlag, Stuttgart (1973)Google Scholar
  19. 19.
    Rechenberg, I.: Evolutionsstrategie 1994. Friedrich Frommann Verlag, Stuttgart (1994)MATHGoogle Scholar
  20. 20.
    Rudolph, G.: Convergence Properties of Evolutionary Algorithms. Verlag Dr.Kovač, Hamburg (1997)Google Scholar
  21. 21.
    Salomon, R.: Evolutionary search and gradient search: Similarities and differences. IEEE Transactions on Evolutionary Computation 2(2), 45–55 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dirk V. Arnold
    • 1
  1. 1.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada

Personalised recommendations