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On the Distribution of EMOA Hypervolumes

  • Olaf Mersmann
  • Heike Trautmann
  • Boris Naujoks
  • Claus Weihs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6073)

Abstract

In recent years, new approaches for multi-modal and multiobjective stochastic optimisation have been developed. It is a rather normal process that these experimental fields develop independently from other scientific areas. However, the connection between stochastic optimisation and statistics is obvious and highly appreciated. Recent works, such as sequential parameter optimisation (SPO, cf. Bartz-Beielstein [1]) or online convergence detection (OCD, cf. Trautmann et al [2]), have combined methods from evolutionary computation and statistics.

One important aspect in statistics is the analysis of stochastic outcomes of experiments and optimization methods, respectively. To this end, the optimisation runs of different evolutionary multi-objective optimisation algorithms (EMOA, cf. Deb [3] or Coello Coello et al. [4]) are treated as experiments to analyse the stochastic behavior of the results and to approximate the distribution of the performance of the EMOA. To combine the outcome of an EMOA and receive a single performance indicator value, the hypervolume (HV) indicator is considered, which is the only known unary quality indicator in this field (cf. Zitzler et al. [5]). The paper at hand investigates and compares the HV indicator outcome of multiple runs of two EMOA on different mathematical test cases.

Keywords

Evolutionary Computation Kernel Density Estimate Function Evaluation Multi Criterion Optimization Parallel Coordinate Plot 
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.
    Bartz-Beielstein, T.: Experimental Research in Evolutionary Computation – The New Experimentalism. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Trautmann, H., Wagner, T., Naujoks, B., Preuss, M., Mehnen, J.: Statistical Methods for Convergence Detection of Multi-Objective Evolutionary Algorithms. Evolutionary Computation Winter 2009 17(4), 493–509 (2009)CrossRefGoogle Scholar
  3. 3.
    Deb, K.: Multi-objective Optimization using Evolutionary Algorithms. Wiley, Chichester (2001)zbMATHGoogle Scholar
  4. 4.
    Coello, C.C., Veldhuizen, D.V., Lamont, G.B.: Evolutionary Algorithms for Solving Multi-Objective Problems, 2nd edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  5. 5.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Trans. on Evolutionary Computation 8(2), 117–132 (2003)CrossRefGoogle Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Trans. on Evolutionary Computation 6(8) (2002)Google Scholar
  7. 7.
    Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)zbMATHCrossRefGoogle Scholar
  8. 8.
    Wagner, T., Beume, N., Naujoks, B.: Pareto-, aggregation-, and indicator-based methods in many-objective optimization. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T., et al. (eds.) EMO 2007. LNCS, vol. 4403, pp. 742–756. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation 8(2), 173–195 (2000)CrossRefGoogle Scholar
  10. 10.
    Mersmann, O., Trautmann, H., Steuer, D.: mco: Multi criteria optimization algorithms and related functions, R package version 1.0.7 (2009), http://cran.r-project.org/web/packages/mco/index.html
  11. 11.
    Mersmann, O., Trautmann, H., Naujoks, B.: emoa: Evolutionary Multiobjective Optimization Algorithms, R package version 0.1-0 (2009), http://www.statistik.tu-dortmund.de/~olafm/emoa/
  12. 12.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation 8(2), 173–195 (2000)CrossRefGoogle Scholar
  13. 13.
    Scott, D.W.: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley Series in Probability and Statistics. Wiley-Interscience, Hoboken (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Wegmann, E.: Hyperdimensional data analysis using parallel coordinates. Journal of the American Statistical Association 85, 664–675 (1990)CrossRefGoogle Scholar
  15. 15.
    Mersmann, O., Trautmann, H., Naujoks, B., Weihs, C.: Benchmarking evolutionary multiobjective optimization algorithms. In: Ishibuchi, H., et al. (eds.) Congress on Evolutionary Computation (CEC). IEEE Press, Piscataway (accepted, 2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Olaf Mersmann
    • 1
  • Heike Trautmann
    • 1
  • Boris Naujoks
    • 2
  • Claus Weihs
    • 1
  1. 1.Statistics FacultyTU Dortmund UniversityGermany
  2. 2.Log!n GmbH, SchwelmGermany

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