On the Behaviour of the (1+1)-ES for a Simple Constrained Problem

  • Dirk V. Arnold
  • Daniel Brauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)

Abstract

This paper studies the behaviour of the (1 + 1)-ES when applied to a linear problem with a single linear constraint. It goes beyond previous work by considering constraint planes that do not contain the gradient direction. The behaviour of the distance of the search point from the constraint plane forms a Markov chain. The limit distribution of that chain is approximated using an exponential model, and progress rates and success probabilities are derived. Consequences for the working of step length adaptation mechanisms based on success probabilities are discussed.

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References

  1. 1.
    Arnold, D.V.: On the use of evolution strategies for optimising certain positive definite quadratic forms. In: Proceedings of the 2007 Genetic and Evolutionary Computation Conference — GECCO 2007, pp. 634–641. ACM Press, New York (2007)Google Scholar
  2. 2.
    Arnold, D.V., MacLeod, A.: Step length adaptation on ridge functions. Evolutionary Computation 16(2), 151–184 (2008)CrossRefGoogle Scholar
  3. 3.
    Bäck, T., Fogel, D.B., Michalewicz, Z.: Handbook of Evolutionary Computation. Oxford University Press, Oxford (1997)CrossRefMATHGoogle Scholar
  4. 4.
    Beyer, H.-G.: Ein Evolutionsverfahren zur mathematischen Modellierung stationärer Zustände in dynamischen Systemen. PhD thesis, Hochschule für Architektur und Bauwesen, Weimar (1989)Google Scholar
  5. 5.
    Beyer, H.-G.: The Theory of Evolution Strategies. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Coello Coello, C.A.: Constraint-handling techniques used with evolutionary algorithms. In: Proceedings of the 2007 Genetic and Evolutionary Computation Conference — GECCO 2007, pp. 3057–3077. ACM Press, New York (2007)Google Scholar
  7. 7.
    Kramer, O., Schwefel, H.-P.: On three new approaches to handle constraints within evolution strategies. Natural Computing 5(4), 363–385 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello Coello, C.A., Deb, K.: Problem definitions and evaluation criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization. Technical report, Nanyang Technological University, Singapore (2006)Google Scholar
  9. 9.
    Meyer-Nieberg, S., Beyer, H.-G.: Why noise be good: Additive noise on the sharp ridge. In: Proceedings of the 2008 Genetic and Evolutionary Computation Conference — GECCO 2008, may, ACM Press, New York (to appear, 2008)Google Scholar
  10. 10.
    Montes, E.M., Coello Coello, C.A.: A simple multi-membered evolution strategy to solve constrained optimization problems. IEEE Transactions on Evolutionary Computation 9(1), 1–17 (2005)CrossRefGoogle Scholar
  11. 11.
    Oyman, A.I., Deb, K., Beyer, H.-G.: An alternative constraint handling method for evolution strategies. In: Proc. of the 1999 IEEE Congress on Evolutionary Computation, pp. 612–619. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  12. 12.
    Rechenberg, I.: Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Friedrich Frommann Verlag (1973)Google Scholar
  13. 13.
    Rechenberg, I.: Evolutionsstrategie ‘94. Friedrich Frommann Verlag (1994)Google Scholar
  14. 14.
    Runarsson, T.P., Yao, X.: Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation 4(3), 274–283 (2000)CrossRefGoogle Scholar
  15. 15.
    Schwefel, H.-P.: Numerical Optimization of Computer Models. Wiley, Chichester (1981)MATHGoogle Scholar
  16. 16.
    Schwefel, H.-P.: Evolution and Optimum Seeking. Wiley, Chichester (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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