Multiobjective Optimization on a Limited Budget of Evaluations Using Model-Assisted \(\mathcal{S}\)-Metric Selection

  • Wolfgang Ponweiser
  • Tobias Wagner
  • Dirk Biermann
  • Markus Vincze
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)


Real-world optimization problems often require the consideration of multiple contradicting objectives. These multiobjective problems are even more challenging when facing a limited budget of evaluations due to expensive experiments or simulations. In these cases, a specific class of multiobjective optimization algorithms (MOOA) has to be applied. This paper provides a review of contemporary multiobjective approaches based on the singleobjective meta-model-assisted ’Efficient Global Optimization’ (EGO) procedure and describes their main concepts. Additionally, a new EGO-based MOOA is introduced, which utilizes the \(\mathcal{S}\)-metric or hypervolume contribution to decide which solution is evaluated next. A benchmark on recently proposed test functions is performed allowing a budget of 130 evaluations. The results point out that the maximization of the hypervolume contribution within a real multiobjective optimization is superior to straightforward adaptations of EGO making our new approach capable of approximating the Pareto front of common problems within the allowed budget of evaluations.


Efficient Global Optimization \(\mathcal{S}\)-metric Design and Analysis of Computer Experiments Multiobjective Optimization Real-World Problems 


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  1. 1.
    Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. Dissertation, Swiss Federal Institute of Technology (ETH), Zürich (1999)Google Scholar
  2. 2.
    Knowles, J.: ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans. Evolutionary Computation 10(1), 50–66 (2006)CrossRefGoogle Scholar
  3. 3.
    Biermann, D., Weinert, K., Wagner, T.: Model-based optimization revisited: Towards real-world processes. In: [29], pp. 2980–2987Google Scholar
  4. 4.
    Jeong, S., Obayashi, S.: Efficient global optimization (EGO) for multi-objective problem and data mining. In: [30], pp. 2138–22145Google Scholar
  5. 5.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. Global Optimization 13(4), 455–492 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Wang, G.G., Shan, S.: Review of metamodeling techniques in support of engineering design optimization. Journal of Mechanical Design 129(4), 370–380 (2007)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Statistical Science 4(4), 409–423 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gonzalez, L.F., Walker, R., Periaux, K.S.,, J.: Multidisciplinary design optimisation of unmanned aerial systems (UAS) using meta model assisted evolutionary algorithms. In: 16th Australasian Fluid Mechanics Conf., pp. 471–474 (2007)Google Scholar
  9. 9.
    D’Angelo, S., Minisci, E.: Multi-objective evolutionary optimization of subsonic airfoils by kriging approximation and evolution control. In: [30], pp. 1262–1267Google Scholar
  10. 10.
    Emmerich, M., Naujoks, B.: Metamodel-assisted multiobjective optimisation strategies and their application in airfoil design. In: Parmee, I.C. (ed.) Adaptive Computing in Design and Manufacture VI, pp. 249–260. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Naujoks, B., Beume, N., Emmerich, M.: Metamodel-assisted SMS-EMOA applied to airfoil optimisation tasks. In: Schilling, R., et al. (eds.) Evolutionary and Deterministic Methods for Design, Optimization and Control with Applications to Industrial and Societal Problems (EUROGEN 2005) (CD-ROM). FLM (2005)Google Scholar
  12. 12.
    Emmerich, M., Giannakoglou, K., Naujoks, B.: Single- and multi-objective evolutionary optimization assisted by gaussian random field metamodels. IEEE Trans. Evolutionary Computation 10(4), 421–439 (2006)CrossRefGoogle Scholar
  13. 13.
    Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA 44, 879–891 (2006)CrossRefGoogle Scholar
  14. 14.
    Sasena, M., Papalambros, P., Goovaerts, P.: Exploration of metamodeling sampling criteria for constrained global optimization. Engineering Optimization 34, 263–278 (2002)CrossRefGoogle Scholar
  15. 15.
    Ponweiser, W., Wagner, T.: Clustered multiple generalized expected improvement: A novel infill sampling criterion for surrogate models. In: [29], pp. 3514–3521 Google Scholar
  16. 16.
    Knowles, J.D., Hughes, E.J.: Multiobjective optimization on a budget of 250 evaluations. In: [31], pp. 176–190 Google Scholar
  17. 17.
    Deb, K., Agrawal, S., Pratab, A., Meyarivan, T.: A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-II. In: Schoenauer, M., et al. (eds.) Proc. 6th Int’l Conf. Parallel Problem Solving from Nature, pp. 849–858. Springer, Heidelberg (2000)Google Scholar
  18. 18.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms – A comparative case study. In: Eiben, A.E., et al. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  19. 19.
    Knowles, J.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. PhD thesis, Department of Computer Science, University of Reading (2002)Google Scholar
  20. 20.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. Trans. Evolutionary Computation 8(2), 117–132 (2003)CrossRefGoogle Scholar
  21. 21.
    Wagner, T., Beume, N., Naujoks, B.: Pareto-, aggregation-, and indicator-based methods in many-objective optimization. In: Obayashi, S., et al. (eds.) Proc. 4th Int’l Conf. Evolutionary Multi-Criterion Optimization, pp. 742–756. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Emmerich, M., Beume, N., Naujoks, B.: An emo algorithm using the hypervolume measure as selection criterion. In: [31], pp. 62–76 (2005)Google Scholar
  23. 23.
    Huang, V.L., Qin, A.K., Deb, K., Zitzler, E., Suganthan, P.N., Liang, J.J., Preuss, M., Huband, S.: Problem definitions for performance assessment of multi-objective optimization algorithms. Technical report, Nanyang Technological University, Singapore (2007)Google Scholar
  24. 24.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. MIT Evolutionary Computation 8(2), 173–195 (2000)CrossRefGoogle Scholar
  25. 25.
    Okabe, T., Jin, Y., Olhofer, M., Sendhoff, B.: On test functions for evolutionary multi-objective optimization. In: Yao, X., et al. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 792–802. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation 9(2), 159–195 (2001)CrossRefGoogle Scholar
  27. 27.
    Fonseca, C., Paquete, L., Lopez-Ibanez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: Congr. Evolutionary Computation, pp. 1157–1163. IEEE, Los Alamitos (2006)Google Scholar
  28. 28.
    Hansen, M.P., Jaszkiewicz, A.: Evaluating the quality of approximations to the non-dominated set. Technical Report IMM1998-7, Technical University of Denmark (1998)Google Scholar
  29. 29.
    Michalewicz, Z., et al. (eds.): Proc. Congr. Evolutionary Computation. IEEE, Los Alamitos (2008)Google Scholar
  30. 30.
    Corne, D., et al. (eds.): Proc. Congr. Evolutionary Computation. IEEE, Los Alamitos (2005)Google Scholar
  31. 31.
    Coello, C.A.C., et al. (eds.): Proc. 3rd Int’l Conf. Evolutionary Multi-Criterion Optimization. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Wolfgang Ponweiser
    • 1
  • Tobias Wagner
    • 2
  • Dirk Biermann
    • 2
  • Markus Vincze
    • 1
  1. 1.Automation and Control InstituteVienna University of TechnologyViennaAustria
  2. 2.Institute of Machining Technology (ISF)Technische UniversitätDortmundGermany

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