Evolutionary search for minimal elements in partially ordered finite sets

  • Günter Rudolph
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1447)

Abstract

The task of finding minimal elements of a partially ordered set is a generalization of the task of finding the global minimum of a real-valued function or of finding Pareto-optimal points of a multicriteria optimization problem. It is shown that evolutionary algorithms are able to converge to the set of minimal elements in finite time with probability one, provided that the search space is finite, the time-invariant variation operator is associated with a positive transition probability function and that the selection operator obeys the so-called ‘elite preservation strategy.’

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. M. Fonseca and P. J. Fleming. An overview of evolutionary algorithms in multiobjective optimization. Evolutionary Computation, 3(1):1–16, 1995.Google Scholar
  2. 2.
    H. Tamaki, H. Kita, and S. Kobayashi. Multi-objective optimization by genetic algorithms: a review. In Proceedings of the 3rd IEEE International Conference on Evolutionary Computation, pages 517–522. IEEE Press, Piscataway (NJ), 1996.Google Scholar
  3. 3.
    J. Horn. Multicriterion decision making. In T. Bäck, D. B. Fogel, and Z. Michalewicz, editors, Handbook of Evolutionary Computation, pages F1.9:1–15. IOP Publishing and Oxford University Press, New York and Bristol (UK), 1997.Google Scholar
  4. 4.
    A. E. Eiben, E. H. L. Aarts, and K. M. Van Hee. Global convergence of genetic algorithms: A markov chain analysis. In H.-P. Schwefel and R. Männer, editors, Parallel Problem Solving from Nature, pages 4–12. Springer, Berlin and Heidelberg, 1991.Google Scholar
  5. 5.
    D. B. Fogel. Asymptotic convergence properties of genetic algorithms and evolutionary programming: Analysis and experiments. Cybernetics and Systems, 25(3):389–407, 1994.Google Scholar
  6. 6.
    G. Rudolph. Convergence properties of canonical genetic algorithms. IEEE Transactions on Neural Networks, 5(1):96–101, 1994.Google Scholar
  7. 7.
    J. Suzuki. A markov chain analysis on simple genetic algorithms. IEEE Transactions on Systems, Man, and Cybernetics, 25(4):655–659, 1995.Google Scholar
  8. 8.
    J. Ester. Systemanalyse und mehrkriterielle Entscheidung. VEB Verlag Technik, Berlin, 1987.Google Scholar
  9. 9.
    W. T. Trotter. Partially ordered sets. In R. L. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics, Vol. 1, pages 433–480. Elsevier Science, Amsterdam, 1995.Google Scholar
  10. 10.
    W. Klingenberg and P. Klein. Lineare Algebra und analytische Geometrie, Band 1. Bibliographisches Institut, Mannheim, 1971.Google Scholar
  11. 11.
    G. Rudolph. Convergence of evolutionary algorithms in general search spaces. In Proceedings of the 3rd IEEE Conference on Evolutionary Computation, pages 50–54. IEEE Press, Piscataway (NJ), 1996.Google Scholar
  12. 12.
    M. Peschel and C. Riedel. Use of vector optimization in multiobjective decision making. In D. E. Bell, R. L. Keeney, and H. Raiffa, editors, Conflicting Objectives in Decisions, pages 97–121. Wiley, Chichester, 1977.Google Scholar
  13. 13.
    J. S. Rosenthal. Convergence rates for markov chains. SIAM Review, 37(3):387–405, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Günter Rudolph
    • 1
  1. 1.Fachbereich InformatikUniversität DortmundDortmundGermany

Personalised recommendations