Advertisement

Journal of Global Optimization

, Volume 55, Issue 2, pp 313–336 | Cite as

Simultaneous kriging-based estimation and optimization of mean response

  • Janis Janusevskis
  • Rodolphe Le Riche
Article

Abstract

Robust optimization is typically based on repeated calls to a deterministic simulation program that aim at both propagating uncertainties and finding optimal design variables. Often in practice, the “simulator” is a computationally intensive software which makes the computational cost one of the principal obstacles to optimization in the presence of uncertainties. This article proposes a new efficient method for minimizing the mean of the objective function. The efficiency stems from the sampling criterion which simultaneously optimizes and propagates uncertainty in the model. Without loss of generality, simulation parameters are divided into two sets, the deterministic optimization variables and the random uncertain parameters. A kriging (Gaussian process regression) model of the simulator is built and a mean process is analytically derived from it. The proposed sampling criterion that yields both optimization and uncertain parameters is the one-step ahead minimum variance of the mean process at the maximizer of the expected improvement. The method is compared with Monte Carlo and kriging-based approaches on analytical test functions in two, four and six dimensions.

Keywords

Kriging based optimization Uncertainty propagation Optimization under uncertainty Robust optimization Gaussian process Expected improvement 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Audze P., Eglajs V.: New approach for design of experiments. Problems Dyn. Strengths 35, 104–107 (1977) (in Russian)Google Scholar
  2. 2.
    Auzins, J.: Direct optimization of experimental designs. In: 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Number AIAA Paper: 2004-4578 (2004)Google Scholar
  3. 3.
    Beyer H.-G., Sendhoff B.: Robust optimization—a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196(33–34), 3190–3218 (2007)CrossRefGoogle Scholar
  4. 4.
    Dellino, G., Kleijnen, J.P.C., Meloni, C.: Robust optimization in simulation: Taguchi and Krige combined. Technical Report (2009)Google Scholar
  5. 5.
    Du X., Chen W.: Towards a better understanding of modeling feasibility robustness in engineering design. ASME J. Mech. Des. 122, 385–394 (1999)CrossRefGoogle Scholar
  6. 6.
    Eldred, M.S., Bichon, B.J., Mahadevan, S.: Reliability-based design optimization using efficient global reliability analysis. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2009-2261Google Scholar
  7. 7.
    Forrester A.I., Bressloff N.W., Keane A.J.: Optimization using surrogate models and partially converged computational fluid dynamics simulations. Proc. R. Soc. A Math. Phys. Eng. Sci. 462(2071), 2177–2204 (2006)CrossRefGoogle Scholar
  8. 8.
    Forrester, A.I.J., Jones, D.R.: Global optimization of deceptive functions with sparse sampling. In: 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 10–12 Sep 2008, 15 pp, Victoria, Canada (2008)Google Scholar
  9. 9.
    Ginsbourger D., Le Riche R., Carraro L.: Kriging is well-suited to parallelize optimization. In: Tenne, Y., Goh, C.-K. (eds) Computational Intelligence in Expensive Optimization Problems, Springer series in Evolutionary Learning and Optimization, pp. 131–162. Springer, Berlin (2009)Google Scholar
  10. 10.
    Girard, A.: Approximate Methods for Propagation of Uncertainty with Gaussian Process Models. PhD thesis (2004)Google Scholar
  11. 11.
    Hansen N.: The CMA evolution strategy: a comparing review. In: Lozano, J.A., Larranaga, P., Inza, I., Bengoetxea, E. (eds) Towards a New Evolutionary Computation. Advances on Estimation of Distribution Aalgorithms, pp. 75–102. Springer, Berlin (2006)CrossRefGoogle Scholar
  12. 12.
    Huang D., Allen T.T., Notz W.I., Zeng N.: Global optimization of stochastic black-box systems via sequential kriging meta-models. J. Global Optim. 34(3), 441–466 (2006)CrossRefGoogle Scholar
  13. 13.
    Huang B., Du X.: A robust design method using variable transformation and Gauss–Hermite integration. Int. J. Numer. Methods Eng. 66(12), 1841–1858 (2006)CrossRefGoogle Scholar
  14. 14.
    Janusevskis, J.: KRIging Scilab Package. http://atoms.scilab.org/toolboxes/krisp/ (2011)
  15. 15.
    Janusevskis, J., Le Riche, R.: Simultaneous kriging-based sampling for optimization and uncertainty propagation. Technical Report, Equipe: Calcul de Risque, Optimisation et Calage par Utilisation de Simulateurs—CROCUS-ENSMSE—UR LSTI—Ecole Nationale Supérieure des Mines de Saint-Etienne. Deliverable no. 2.2.2-A of the ANR/OMD2 project available as http://hal.archives-ouvertes.fr/hal-00506957
  16. 16.
    Jin, R., Du, X., Chen, W.: The use of metamodeling techniques for optimization under uncertainty. In: 2001 ASME Design Automation Conference, Paper No. DAC21039, pp. 99–116. (2001)Google Scholar
  17. 17.
    Jones D.R.: A taxonomy of global optimization methods based on response surfaces. J. Global Optim. 21, 345–383 (2001)CrossRefGoogle Scholar
  18. 18.
    Jones D.R., Schonlau M., Welch W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)CrossRefGoogle Scholar
  19. 19.
    Le Riche, R., Picheny, V.: Gears design with shape uncertainties using controlled monte carlo simulations and kriging. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2009-2257Google Scholar
  20. 20.
    McKay M.D., Beckman R.J., Conover W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)Google Scholar
  21. 21.
    O’Hagan A.: Bayes-hermite quadrature. J. Stat. Plan. Inference 29, 245–260 (1991)CrossRefGoogle Scholar
  22. 22.
    Park G.-J., Lee T.-H., Kwon H.L., Hwang K.-H.: Robust design : an overview. AIAA J. 44(1), 181–191 (2006)CrossRefGoogle Scholar
  23. 23.
    Picheny, V., Ginsbourger, D., Richet, Y.: Noisy expected improvement and on-line computation time allocation for the optimization of simulators with tunable fidelity. In: EngOpt 2010—2nd International Conference on Engineering OptimizationGoogle Scholar
  24. 24.
    Rasmussen C.E., Williams C.K.I.: Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, Cambridge (2005)Google Scholar
  25. 25.
    Rasmussen, C.E., Ghahramani, Z.: Bayesian monte carlo. In: Thrun, S., Becker, S., Obermayer, K. (eds.) Advances in Neural Information Processing Systems, vol. 15. MIT Press, CambridgeGoogle Scholar
  26. 26.
    Sacks J., Welch W.J., Mitchell T.J., Wynn H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)CrossRefGoogle Scholar
  27. 27.
    Sahinidis N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28, 971–983 (2004)CrossRefGoogle Scholar
  28. 28.
    Salazar D., Le Riche R., Bay X.: An empirical study of the use of confidence levels in RBDO with Monte Carlo simulations. In: Piotr, Breitkopf., Rajan Filomeno, Coelho (eds) Multidisciplinary Design Optimization in Computational Mechanics, Wiley, New York (2009)Google Scholar
  29. 29.
    Sasena, M.J.: Flexibility and Efficiency Enhancements for Constrained Global Design Optimization with Kriging Approximations. PhD thesis (2002)Google Scholar
  30. 30.
    Taflanidis, A.: Stochastic System Design and Applications to Stochastically Robust Structural Control. PhD thesis (2007)Google Scholar
  31. 31.
    Vazquez E., Villemonteix J., Sidorkiewicz M., Walter E.: Global optimization based on noisy evaluations: an empirical study of two statistical approaches. J. Phys. Conf. Ser. 135(N012100), 8 (2008)Google Scholar
  32. 32.
    Villemonteix J., Vázquez E., Walter E.: An informational approach to the global optimization of expensive-to-evaluate functions. J. Global Optim. 44(4), 509–534 (2009)CrossRefGoogle Scholar
  33. 33.
    Williams B.J., Santner T.J., Notz W.I.: Sequential design of computer experiments to minimize integrated response functions. Stat. Sin 10(4), 1133–1152 (2000)Google Scholar
  34. 34.
    Wojkiewicz, S.F., Trucano, T., Eldred, M.S., Giunta, A.A.: Formulations for surrogate-based optimization under uncertainty. In: 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA Paper 2002-5585Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.H. Fayol InstituteEcole Nationale Supérieure des Mines de Saint-EtienneSaint-ÉtienneFrance
  2. 2.CNRS UMR 5146, H. Fayol InstituteEcole Nationale Supérieure des Mines de Saint-EtienneSaint-ÉtienneFrance

Personalised recommendations