On Expected-Improvement Criteria for Model-based Multi-objective Optimization

  • Tobias Wagner
  • Michael Emmerich
  • André Deutz
  • Wolfgang Ponweiser
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6238)

Abstract

Surrogate models, as used for the Design and Analysis of Computer Experiments (DACE), can significantly reduce the resources necessary in cases of expensive evaluations. They provide a prediction of the objective and of the corresponding uncertainty, which can then be combined to a figure of merit for a sequential optimization. In single-objective optimization, the expected improvement (EI) has proven to provide a combination that balances successfully between local and global search. Thus, it has recently been adapted to evolutionary multi-objective optimization (EMO) in different ways. In this paper, we provide an overview of the existing EI extensions for EMO and propose new formulations of the EI based on the hypervolume. We set up a list of necessary and desirable properties, which is used to reveal the strengths and weaknesses of the criteria by both theoretical and experimental analyses.

Keywords

Design and Analysis of Computer Experiments Expected Improvement Hypervolume Indicator Multi-Objective Optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Knowles, J., Nakayama, H.: Meta-modeling in multiobjective optimization. In: Branke, J., et al. (eds.) Multiobjective Optimization – Interactive and Evolutionary Approaches, pp. 461–478. Springer, Berlin (2008)Google Scholar
  2. 2.
    Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Mockus, J.B., Tiesis, V., Zilinskas, A.: The application of bayesian methods for seeking the extremum. In: Dixon, L.C.W., Szegö, G.P. (eds.) Towards Global Optimization, vol. 2, pp. 117–129. Amsterdam, New York (1978)Google Scholar
  4. 4.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bartz-Beielstein, T., Lasarczyk, C., Preuss, M.: Sequential parameter optimization. In: McKay, B., et al. (eds.) Proc. CEC, pp. 773–780. IEEE, Los Alamitos (2005)Google Scholar
  6. 6.
    Zitzler, E., Thiele, L., Bader, J.: On set-based multiobjective optimization. IEEE Trans. Evol. Comput. 14(1), 58–79 (2010)CrossRefGoogle Scholar
  7. 7.
    Emmerich, M.: Single- and Multi-objective Evolutionary Design Optimization Assisted by Gaussian Random Field Metamodels. PhD thesis, Universität Dortmund (2005)Google Scholar
  8. 8.
    Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44(4), 879–891 (2006)CrossRefGoogle Scholar
  9. 9.
    Liu, W., Zhang, Q., Tsang, E., Liu, C., Virginas, B.: On the performance of metamodel assisted MOEA/D. In: Kang, L., Liu, Y., Zeng, S., et al. (eds.) ISICA 2007. LNCS, vol. 4683, pp. 547–557. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Emmerich, M., Deutz, A.H., Klinkenberg, J.W.: The computation of the expected improvement in dominated hypervolume of pareto front approximations. Technical Report 4-2008 Leiden Institute of Advanced Computer Science, LIACS (2008), http://www.liacs.nl/~emmerich/TR-ExI.pdf
  11. 11.
    Zhang, Q., Liu, W., Tsang, E., Virginas, B.: Expensive multiobjective optimization by MOEA/D with gaussian process model. IEEE Trans. Evol. Comput. (2010); Early Access (will be published)Google Scholar
  12. 12.
    Knowles, J.: ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans. Evol. Comput. 10(1), 50–66 (2006)CrossRefGoogle Scholar
  13. 13.
    Jeong, S., Obayashi, S.: Efficient global optimization (EGO) for multi-objective problem and data mining. In: Corne, D., et al. (eds.) Proc. CEC, pp. 2138–2145. IEEE, Los Alamitos (2005)Google Scholar
  14. 14.
    Ponweiser, W., Wagner, T., Biermann, D., Vincze, M.: Multiobjective optimization on a limited amount of evaluations using model-assisted \(\mathcal{S}\)-metric selection. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 784–794. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)CrossRefGoogle Scholar
  16. 16.
    Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York (1986)MATHGoogle Scholar
  17. 17.
    Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: Optimal μ-distributions and the choice of the reference point. In: Garibay, I., et al. (eds.) Proc. FOGA, pp. 87–102. ACM, New York (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tobias Wagner
    • 1
  • Michael Emmerich
    • 2
  • André Deutz
    • 2
  • Wolfgang Ponweiser
    • 3
  1. 1.Institute of Machining Technology (ISF)Technische Universität DortmundDortmundGermany
  2. 2.Leiden Institute of Advanced Computer Science (LIACS)Universiteit LeidenLeidenThe Netherlands
  3. 3.Automation and Control InstituteVienna University of TechnologyViennaAustria

Personalised recommendations