Exploring the Performance of Stochastic Multiobjective Optimisers with the Second-Order Attainment Function

  • Carlos M. Fonseca
  • Viviane Grunert da Fonseca
  • Luís Paquete
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3410)


The attainment function has been proposed as a measure of the statistical performance of stochastic multiobjective optimisers which encompasses both the quality of individual non-dominated solutions in objective space and their spread along the trade-off surface. It has also been related to results from random closed-set theory, and cast as a mean-like, first-order moment measure of the outcomes of multiobjective optimisers. In this work, the use of more informative, second-order moment measures for the evaluation and comparison of multiobjective optimiser performance is explored experimentally, with emphasis on the interpretability of the results.


Covariance Function Objective Space Moment Measure Multiobjective Genetic Algorithm Empirical Covariance Function 
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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  1. 1.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transactions on Evolutionary Computation 7, 117–132 (2003)CrossRefGoogle Scholar
  2. 2.
    Goutsias, J.: Modeling random shapes: An introduction to random closed set theory. In: Haralick, R.M. (ed.) Mathematical Morphology: Theory and Hardware. Oxford Series in Optical & Imaging Sciences. Oxford University Press, New York (1996)Google Scholar
  3. 3.
    Grunert da Fonseca, V., Fonseca, C.M., Hall, A.O.: Inferential performance assessment of stochastic optimisers and the attainment function. In: Zitzler, E., Deb, K., Thiele, L., Coello Coello, C.A., Corne, D. (eds.) EMO 2001. LNCS, vol. 1993, p. 213. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Justel, A., Peña, D., Zamar, R.: A multivariate Kolmogorov-Smirnov test of goodness of fit. Statistics and Probability Letters 35, 251–259 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fonseca, C.M., Fleming, P.J.: On the performance assessment and comparison of stochastic multiobjective optimizers. In: Ebeling, W., Rechenberg, I., Voigt, H.-M., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 584–593. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  6. 6.
    Shaw, K.J., Nortcliffe, A.L., Thompson, M., Love, J., Fleming, P.: Assessing the performance of multiobjective genetic algorithms for optimization of a batch process scheduling problem. In: Proceedings of the Congress on Evolutionary Computation (CEC1999), Washington DC, vol. 1, pp. 37–45 (1999)Google Scholar
  7. 7.
    Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications, 2nd edn. Wiley Series in Probability and Statistics. Wiley & Sons, Chichester (1995)MATHGoogle Scholar
  8. 8.
    Good, P.I.: Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses, 2nd edn. Springer Series in Statistics. Springer, New York (2000)MATHGoogle Scholar
  9. 9.
    Bickel, P.J.: A distribution free version of the Smirnov two sample test in the p-variate case. The Annals of Mathematical Statistics 40, 1–23 (1969)MATHCrossRefGoogle Scholar
  10. 10.
    Barratt, C., Boyd, S.: Example of exact trade-offs in linear control design. IEEE Control Systems Magazine 9, 46–52 (1989)CrossRefGoogle Scholar
  11. 11.
    Fonseca, C.M., Fleming, P.J.: Multiobjective optimal controller design with genetic algorithms. In: Proc. IEE Control 1994 International Conference, Warwick, U.K, vol. 1, pp. 745–749 (1994)Google Scholar
  12. 12.
    Fonseca, C.M., Fleming, P.J.: Multiobjective genetic algorithms made easy: selection, sharing and mating restriction. In: First IEE/IEEE International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications, Sheffield, U.K., pp. 45–52 (1995)Google Scholar
  13. 13.
    Paquete, L., Chiarandini, M., Stützle, T.: Pareto local optimum sets in the biobjective traveling salesman problem: An experimental study. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation. Lecture Notes in Economics and Mathematical Systems, vol. 535, pp. 177–200. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Bentley, J.: Fast algorithms for geometric traveling salesman problems. ORSA Journal on Computing 4, 387–411 (1992)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Carlos M. Fonseca
    • 1
  • Viviane Grunert da Fonseca
    • 1
    • 2
  • Luís Paquete
    • 3
    • 4
  1. 1.CSI – Centro de Sistemas Inteligentes, Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal
  2. 2.INUAF – Instituto Superior D. Afonso IIILouléPortugal
  3. 3.Fachgebiet Intellektik, Fachbereich InformatikTechnische Universität DarmstadtDarmstadtGermany
  4. 4.Faculdade de EconomiaUniversidade do AlgarveFaroPortugal

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