Structural and Multidisciplinary Optimization

, Volume 54, Issue 6, pp 1403–1421 | Cite as

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

  • Maliki Moustapha
  • Bruno SudretEmail author
  • Jean-Marc Bourinet
  • Benoît Guillaume


Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.


Quantile-based design optimization RBDO Kriging Adaptive design of experiments 


  1. Agarwal H, Mozumder CK, Renaud JE, Watson LT (2007) An inverse-measure-based unilevel architecture for reliability-based design optimization. Struct Multidiscip Optim 33(3):217–227CrossRefGoogle Scholar
  2. Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41(2):277–294MathSciNetCrossRefzbMATHGoogle Scholar
  3. Arnold DV, Hansen N (2012) A (1 + 1)-CMA-ES for constrained optimisation. In: Soule T, Moore JH (eds) Genetic and evolutionary computation conference, pp 297–304Google Scholar
  4. Asmussen S, Glynn PW (2007) Stochastic simulation: algorithms and analysis. In: Stochastic modelling and applied probability. Springer, New YorkzbMATHGoogle Scholar
  5. Au S-K (2005) Reliability-based design sensitivity by efficient simulation. Comput Struct 83(14):1048–1061CrossRefGoogle Scholar
  6. Au S-K, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Struct Saf 21(2):135–158CrossRefGoogle Scholar
  7. Au S-K, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16:263–277CrossRefGoogle Scholar
  8. Audet C, Booker AJ, Dennis J, Frank PD, Moore DW (2000) A surrogate-model-based method for constrained optimization. In: Proceedings of the 8th symposium on multidisciplinary analysis and optimization. Long BeachGoogle Scholar
  9. Balesdent M, Morio J, Marzat J (2013) Kriging-based adaptive importance sampling algorithms for rare event estimation. Struct Saf 44:1–10CrossRefGoogle Scholar
  10. Baudoui V (2012) Optimisation robuste multiobjectifs par modèles de substitution. Ph.D thesis, Institut Supérieur de l’Aéronautique et de l’Espace, ToulouseGoogle Scholar
  11. Beyer H-G, Sendhoff B (2007) Robust optimization: a comprehensive survey. Comput Methods Appl Mech Eng 196(33–34): 3190–3218MathSciNetCrossRefzbMATHGoogle Scholar
  12. Bichon B, Eldred M, Swiler L, Mahadevan S, McFarland J (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459–2468CrossRefGoogle Scholar
  13. Blatman G (2009) Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis. Ph.D. thesis, Université Blaise Pascal, Clermont-FerrandGoogle Scholar
  14. Blatman G, Sudret B (2010) Reliability analysis of a pressurized water reactor vessel using sparse polynomial chaos expansions. In: Straub D, Esteva L, Faber M (eds) Proceedings of the 15th IFIP WG7.5 conference on reliability and optimization of structural systems, Munich, Germany. Taylor & Francis, pp 9–16Google Scholar
  15. Bourinet J-M, Deheeger F, Lemaire M (2011) Assessing small failure probabilities by combined subset simulation and support vector machines. Struct Saf 33(6):343–353CrossRefGoogle Scholar
  16. Chateauneuf A, Aoues Y (2008) Structural design optimization considering uncertaintiesm, chapter 9. Taylor & Francis, pp. 217–246Google Scholar
  17. Chen Z, Peng S, Li X, Qiu H, Xiong H, Gao L, Li P (2015) An important boundary sampling method for reliability-based design optimization using Kriging model. Struct Multidiscip Optim 52(1):55–70MathSciNetCrossRefGoogle Scholar
  18. Couckyut I, Dhane T, Demeester P (2013) ooDace toolbox A Matlab Kriging toolbox: getting started. Universiteit GentGoogle Scholar
  19. Deheeger F, Lemaire M (2007) Support vector machine for efficient subset simulations: 2SMART method. In: Proceedings of the 10th international conference on applications of stat. and prob. in civil engineering (ICASP10), Tokyo, JapanGoogle Scholar
  20. Ditlevsen O, Madsen H (1996) Structural reliability methods, Wiley, New YorkGoogle Scholar
  21. Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126(2):225–233CrossRefGoogle Scholar
  22. Dubourg V (2011) Adaptive surrogate models for reliability analysis and reliability-based design optimization. Ph.D. thesis, Université Blaise Pascal, Clermont-FerrandGoogle Scholar
  23. Dubourg V, Sudret B, Bourinet J-M (2011) Reliability-based design optimization using Kriging and subset simulation. Struct Multidiscip Optim 44(5):673–690CrossRefGoogle Scholar
  24. Echard B, Gayton N, Lemaire M. (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct Saf 33(2):145–154CrossRefGoogle Scholar
  25. Enevoldsen I, Sorensen J (1994) Reliability-based optimization in structural engineering. Struct Saf 15 (3):169–196CrossRefGoogle Scholar
  26. Fauriat W, Gayton N (2014) AK-SYS: an adaptation of the AK-MCS method for system reliability. Reliab Eng Syst Saf 123:137–144CrossRefGoogle Scholar
  27. Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div-ASCE 100(1):111–121Google Scholar
  28. Hu C, Youn B (2011) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidiscip Optim 43(3):419–442MathSciNetCrossRefzbMATHGoogle Scholar
  29. Hurtado JE, Alvarez DA (2001) Neural-network-based reliability analysis: a comparative study. Comput Methods Appl Mech 191:113–132CrossRefzbMATHGoogle Scholar
  30. Janusevskis J, Le Riche R (2013) Simultaneous Kriging-based estimation and optimization of mean response. J Glob Optim 55(2):313–336MathSciNetCrossRefzbMATHGoogle Scholar
  31. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Global Optim 13(4): 455–492MathSciNetCrossRefzbMATHGoogle Scholar
  32. Kharmanda G, Mohamed A, Lemaire M (2002) Efficient reliability-based design optimization using a hybrid space with application to finite element analysis. Struct Multidiscip Optim 24(3)Google Scholar
  33. Koehler JR, Owen A (1996) Computer experiments. In: Ghosh S, Rao C (eds) Handbook of statistics, vol 13. North Holland, Amsterdam, pp 261–308Google Scholar
  34. Kuschel N., Rackwitz R. (1997) Two basic problems in reliability-based structural optimization. Math Methods Oper Res 46(3):309–333MathSciNetCrossRefzbMATHGoogle Scholar
  35. Lataniotis C, Marelli S, Sudret B (2015) UQLab user manual – Kriging. Technical report, Chair of risk, safety & uncertainty quantification, ETH Zurich Report #UQLab-V0.9-105Google Scholar
  36. Lee I, Choi KK, Zhao L (2011) Sampling-based RBDO using the stochastic sensitivity analysis and dynamic Kriging method. Struct Multidiscip Optim 44(3):299–317MathSciNetCrossRefzbMATHGoogle Scholar
  37. Lemaire M (2007) Structural reliability. ISTE/Hermes Science PublishingGoogle Scholar
  38. Li X, Qiu H, Chen Z, Gao L, Shao X (2016) A local Kriging approximation method using MPP for reliability-based design optimization. Comput Struct (162): 102–115Google Scholar
  39. Liang J, Mourelatos Z, Tu J (2004) A single-loop method for reliability-based design optimization. In: Proc. DETC’04 ASME 2004 design engineering technical conferences and computers and information in engineering conference, Sept. 28–Oct. 2, 2004, Salt Lake City, Utah, USAGoogle Scholar
  40. Madsen H, Krenk S, Lind N (1986) Methods of structural safety. Prentice Hall, Inc., Englewood CliffsGoogle Scholar
  41. Marelli S, Sudret B (2014) UQLab: a framework for uncertainty quantification in Matlab. In: Vulnerability, uncertainty and risk, proceedings of the 2nd international conference on vulnerability, risk and analysis management (ICVRAM2014), Liverpool, pp 2554–2563Google Scholar
  42. Melchers R (1989) Importance sampling in structural systems. Struct Saf 6:3–10CrossRefGoogle Scholar
  43. Picheny V, Kim NH, Haftka RT, Queipo NV (2008) Conservative predictions using surrogate modeling. In: Proceedings of the 49th AIAA/AME/ASCE/AHS/ASC structures, structural dynamics and materials, 7–10 April 2008, SchaumburgGoogle Scholar
  44. Picheny V, Ginsbourger D, Roustant O, Haftka RT, Kim NH (2010) Adaptive designs of experiments for accurate approximation of a target region. J Mech Des 132(7)Google Scholar
  45. Ranjan P, Bingham D, Michailidis G (2008) Sequential experiment design for contour estimation from complex computer codes. Technometrics 50(4):527–541MathSciNetCrossRefGoogle Scholar
  46. Roustant O, Ginsbourger D, Deville Y (2012) DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J Stat Softw 51(1):1–55CrossRefGoogle Scholar
  47. Santner T, Williams B, Notz W (2003) The design and analysis of computer experiments. Springer, New YorkCrossRefzbMATHGoogle Scholar
  48. Schöbi R, Sudret B (2014) PC-Kriging: a new meta-modelling method and its applications to quantile estimation. In: Li J, Zhao Y (eds) Proceedings of the 17th IFIP WG7.5 conference on reliability and optimization of structural systems, Huangshan, China. Taylor & Francis, New YorkGoogle Scholar
  49. Schöbi R, Sudret B, Marelli S (2016) Rare event estimation using Polynomial-Chaos-Kriging. ASCE-ASME J Risk Uncertainty Eng Syst Part A: Civ Eng D4016002Google Scholar
  50. Schonlau M, Welch WJ, Jones DR (1998) Global versus local search in constrained optimization of computer models. In: New developments and applications in experimental design, vol 34. New developments and applications in experimental design, pp 11–25Google Scholar
  51. Shan S, Wang GG (2008) Reliable design space and complete single-loop reliability-based design optimization. Reliab Eng Syst Saf 93(8):1218–1230CrossRefGoogle Scholar
  52. Taflanidis A, Beck JL (2008) Stochastic subset optimization for optimal reliability problems. Probab Eng Mech 23:324–338CrossRefGoogle Scholar
  53. Trosset M (1997) Taguchi and robust optimization. Technical report, Rice University, HoustonGoogle Scholar
  54. Tu J, Choi KK (1997) A performance measure approach in reliability-based structural optimization. Technical report, Center for computer-aided design. The University of Iowa, IowaGoogle Scholar
  55. Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. J Mech Des 121:557–564CrossRefGoogle Scholar
  56. Viana FAC, Picheny V, Haftka RT (2010) Using cross validation to design conservative surrogates. AIAA J 48(10):2286–2298CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Maliki Moustapha
    • 1
  • Bruno Sudret
    • 2
    Email author
  • Jean-Marc Bourinet
    • 3
  • Benoît Guillaume
    • 4
  1. 1.Institut Pascal, CNRS UMR 6602 & PSA Peugeot CitroënAubièreFrance
  2. 2.Chair of Risk, Safety & Uncertainty QuantificationETH ZurichZurichSwitzerland
  3. 3.Sigma Clermont, Institut Pascal, CNRS UMR 6602AubièreFrance
  4. 4.PSA Peugeot Citroën, Centre Technique de VélizyVélizy-VillacoublayFrance

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