Advertisement

The Relationship between the Covered Fraction, Completeness and Hypervolume Indicators

  • Viviane Grunert da Fonseca
  • Carlos M. Fonseca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7401)

Abstract

This paper investigates the relationship between the covered fraction, completeness, and (weighted) hypervolume indicators for assessing the quality of the Pareto-front approximations produced by multiobjective optimizers. It is shown that these unary quality indicators are all, by definition, weighted Hausdorff measures of the intersection of the region attained by such an optimizer outcome in objective space with some reference set. Moreover, when the optimizer is stochastic, the indicators considered lead to real-valued random variables following particular probability distributions. Expressions for the expected value of these distributions are derived, and shown to be directly related to the first-order attainment function.

Keywords

stochastic multiobjective optimizer performance assessment covered fraction indicator completeness indicator (weighted) hypervolume indicator attainment function expected value Hausdorff measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alberti, G.: Geometric measure theory. In: Françoise, J.P., et al. (eds.) Encyclopedia of Mathematical Physics, vol. 2, pp. 520–527. Elsevier, Oxford (2006)CrossRefGoogle Scholar
  2. 2.
    Bader, J.M.: Hypervolume-Based Search for Multiobjective Optimization: Theory and Methods. Ph.D. thesis, Swiss Federal Institute of Technology, Zurich (2009)Google Scholar
  3. 3.
    Barenblatt, G.I.: Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  4. 4.
    DiBenedetto, E.: Real Analysis. Birkhäuser, Boston (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Fonseca, C.M., Grunert da Fonseca, V., Paquete, L.: Exploring the Performance of Stochastic Multiobjective Optimisers with the Second-Order Attainment Function. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 250–264. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Fonseca, C.M., Guerreiro, A.P., López-Ibáñez, M., Paquete, L.: On the Computation of the Empirical Attainment Function. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 106–120. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Grunert da Fonseca, V., Fonseca, C.M.: The attainment-function approach to stochastic multiobjective optimizer assessment and comparison. In: Bartz-Beielstein, T., et al. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, ch. 5, pp. 103–130. Springer, Berlin (2010)CrossRefGoogle Scholar
  8. 8.
    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, pp. 213–225. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Ito, K. (ed.): Encyclopedic Dictionary of Mathematics 2. The Mathematical Society of Japan. The MIT Press (1987)Google Scholar
  10. 10.
    Lotov, A.V., Bushenkov, V.A., Kamenev, G.K.: Interactive Decision Maps: Approximation and Visualization of Pareto Frontier. Kluwer Academic Publishers, Dordrecht (2004)zbMATHGoogle Scholar
  11. 11.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005)zbMATHGoogle Scholar
  12. 12.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Ulungu, E.L., Teghem, J., Fortemps, P.H., Tuyttens, D.: MOSA method: A tool for solving multiobjective combinatorial optimization problems. Journal of Multi-Criteria Decision Analysis 8(4), 221–236 (1999)zbMATHCrossRefGoogle Scholar
  14. 14.
    Zitzler, E., Brockhoff, D., Thiele, L.: The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Zitzler, E., Knowles, J., Thiele, L.: Quality Assessment of Pareto Set Approximations. In: Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.) Multiobjective Optimization. LNCS, vol. 5252, pp. 373–404. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Zitzler, E., Thiele, L.: Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    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(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Viviane Grunert da Fonseca
    • 1
    • 3
  • Carlos M. Fonseca
    • 2
    • 3
  1. 1.INUAF – Instituto Superior D. Afonso IIILouléPortugal
  2. 2.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal
  3. 3.CEG-IST – Center for Management StudiesInstituto Superior TécnicoLisbonPortugal

Personalised recommendations