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A sequential constraints updating approach for Kriging surrogate model-assisted engineering optimization design problem

  • Jiachang Qian
  • Jiaxiang Yi
  • Yuansheng Cheng
  • Jun Liu
  • Qi ZhouEmail author
Original Article
  • 62 Downloads

Abstract

Kriging surrogate model has been widely used in engineering design optimization problems to replace computational cost simulations. To facilitate the usage of the Kriging surrogate model-assisted engineering optimization design, there are still challenging issues on the updating of Kriging surrogate model for the constraints, since there exists prediction error between the Kriging surrogate model and the real constraints. Ignoring the interpolation uncertainties from the Kriging surrogate model of constraints may lead to infeasible optimal solutions. In this paper, general sequential constraints updating approach based on the confidence intervals from the Kriging surrogate model (SCU-CI) are proposed. In the proposed SCU-CI approach, an objective switching and sequential updating strategy is introduced based on whether the feasibility status of the design alternatives would be changed because of the interpolation uncertainty from the Kriging surrogate model or not. To demonstrate the effectiveness of the proposed SCU-CI approach, nine numerical examples and two practical engineering cases are used. The comparisons between the proposed approach and five existing approaches considering the quality of the obtained optimum and computational efficiency are made. Results illustrate that the proposed SCU-CI approach can generally ensure the feasibility of the optimal solution under a reasonable computational cost.

Keywords

Kriging surrogate model Sequential constraint updating Prediction interval Feasibility 

Notes

Acknowledgements

This work has been supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51805179, 51775203, and the Research Funds of the Maritime Defense Technologies Innovation, and the Research Funds of the defense technologies leadership.

References

  1. 1.
    Hu Z, Mahadevan S (2017) A surrogate modeling approach for reliability analysis of a multidisciplinary system with spatio–temporal output. Struct Multidiscip Optim 56(3):553–569MathSciNetCrossRefGoogle Scholar
  2. 2.
    Jiang C, Qiu H, Yang Z, Chen L, Gao L, Li P (2019) A general failure-pursuing sampling framework for surrogate-based reliability analysis. Reliab Eng Syst Saf 183:47–59CrossRefGoogle Scholar
  3. 3.
    Han Z-H, Zhang Y, Song C-X, Zhang K-S (2017) Weighted gradient-enhanced Kriging for high-dimensional surrogate modeling and design optimization. AIAA J 55(12):4330–4346CrossRefGoogle Scholar
  4. 4.
    Hu J, Zhou Q, Jiang P, Shao X, Xie T (2018) An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical Kriging. Eng Optim 50(1):145–163CrossRefGoogle Scholar
  5. 5.
    Wang H, Chen L, Li E (2017) Time dependent sheet metal forming optimization by using Gaussian process assisted firefly algorithm. Int J Mater Form pp. 1–17Google Scholar
  6. 6.
    Song X, Sun G, Li G, Gao W, Li Q (2012) Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models. Struct Multidiscip Optim 47(2):221–231MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhou Q, Jiang P, Shao X, Hu J, Cao L, Wan L (2017) A variable fidelity information fusion method based on radial basis function. Adv Eng Inform 32:26–39CrossRefGoogle Scholar
  8. 8.
    Bellary SAI, Samad A, Couckuyt I, Dhaene T (2015) A comparative study of kriging variants for the optimization of a turbomachinery system. Eng Comput 32(1):49–59CrossRefGoogle Scholar
  9. 9.
    Jiang P, Zhang Y, Zhou Q, Shao X, Hu J, Shu L (2018) An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS. Appl Intell 48(6):1644–1656CrossRefGoogle Scholar
  10. 10.
    Bouhlel MA, Martins JRRA (2018) Gradient-enhanced kriging for high-dimensional problems. Eng Comput 35(1):157–173CrossRefGoogle Scholar
  11. 11.
    Dong H, Song B, Dong Z, Wang P (2018) SCGOSR: surrogate-based constrained global optimization using space reduction. Appl Soft Comput 65:462–477CrossRefGoogle Scholar
  12. 12.
    Clarke SM, Griebsch JH, Simpson TW (2005) Analysis of support vector regression for approximation of complex engineering analyses. J Mech Des 127(6):1077CrossRefGoogle Scholar
  13. 13.
    Zhou Q, Shao XY, Jiang P, Gao ZM, Zhou H, Shu LS (2016) An active learning variable-fidelity metamodelling approach based on ensemble of metamodels and objective-oriented sequential sampling. J Eng Des 27(4–6):205–231CrossRefGoogle Scholar
  14. 14.
    Jiang C, Cai X, Qiu H, Gao L, Li P (2018) A two-stage support vector regression assisted sequential sampling approach for global metamodeling. Struct Multidiscip Optim 58(4):1657–1672CrossRefGoogle Scholar
  15. 15.
    Chatterjee T, Chakraborty S, Chowdhury R (2017) A critical review of surrogate assisted robust design optimization. Arch Comput Methods Eng 26:245–725MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhou Q, Shao XY, Jiang P, Zhou H, Cao LC, Zhang L (2015) A deterministic robust optimisation method under interval uncertainty based on the reverse model. J Eng Des 26(10–12):416–444CrossRefGoogle Scholar
  17. 17.
    Assari P, Dehghan M (2017) The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision. Eng Comput 33(4):853–870CrossRefGoogle Scholar
  18. 18.
    Zhou Q, Wang Y, Choi S-K, Jiang P, Shao X, Hu J, Shu L (2018) A robust optimization approach based on multi-fidelity metamodel. Struct Multidisciplin Optimization 57(2):775–797CrossRefGoogle Scholar
  19. 19.
    Kaintura A, Spina D, Couckuyt I, Knockaert L, Bogaerts W, Dhaene T (2017) A Kriging and stochastic collocation ensemble for uncertainty quantification in engineering applications. Eng Comput 33:935–949CrossRefGoogle Scholar
  20. 20.
    Han Z, Zimmerman R, Görtz S (2012) Alternative cokriging method for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210CrossRefGoogle Scholar
  21. 21.
    Huang C, Radi B, El Hami A, Bai H (2018) CMA evolution strategy assisted by kriging model and approximate ranking. Appl Intell 48:4288–4304CrossRefGoogle Scholar
  22. 22.
    Shao W, Deng H, Ma Y, Wei Z (2011) Extended Gaussian Kriging for computer experiments in engineering design. Eng Comput 28(2):161–178CrossRefGoogle Scholar
  23. 23.
    Toal DJJ (2015) A study into the potential of GPUs for the efficient construction and evaluation of Kriging models. Eng Comput 32(3):377–404CrossRefGoogle Scholar
  24. 24.
    Cheng J, Jiang P, Zhou Q, Jiexiang H, Tao Y, Leshi S, Xinyu S (2019) A lower confidence bounding approach based on the coefficient of variation for expensive global design optimization. Eng Comput 1:2.  https://doi.org/10.1108/EC-08-2018-0390 Google Scholar
  25. 25.
    Zheng J, Li Z, Gao L, Jiang G, Owen D (2016) A parameterized lower confidence bounding scheme for adaptive metamodel-based design optimization. Eng Comput 33(7):2165–2184CrossRefGoogle Scholar
  26. 26.
    Chen S, Jiang Z, Yang S, Chen W (2016) Multimodel fusion based sequential optimization. AIAA J 55(1):241–254CrossRefGoogle Scholar
  27. 27.
    Regis RG, Shoemaker CA (2007) A stochastic radial basis function method for the global optimization of expensive functions. Informs J Comput 19(4):497–509MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hennig P, Schuler CJ (2012) Entropy search for information-efficient global optimization. J Mach Learn Res 13:1809–1837MathSciNetzbMATHGoogle Scholar
  29. 29.
    Krause A, Ong CS (2011) Contextual gaussian process bandit optimization. Adv Neural Inform Process SystGoogle Scholar
  30. 30.
    Viana FA, Haftka RT, Watson LT (2013) Efficient global optimization algorithm assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689CrossRefzbMATHGoogle Scholar
  31. 31.
    Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sóbester A, Leary SJ, Keane AJ (2005) On the design of optimization strategies based on global response surface approximation models. J Glob Optim 33(1):31–59MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang H, Li E, Li GY (2009) The least square support vector regression coupled with parallel sampling scheme metamodeling technique and application in sheet forming optimization. Mater Des 30(5):1468–1479MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhan D, Qian J, Cheng Y (2017) Balancing global and local search in parallel efficient global optimization algorithms. J Glob Optim 67(4):873–892MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Dong H, Song B, Wang P, Dong Z (2018) Hybrid surrogate-based optimization using space reduction (HSOSR) for expensive black-box functions. Appl Soft Comput 64:641–655CrossRefGoogle Scholar
  36. 36.
    Haftka RT, Villanueva D, Chaudhuri A (2016) Parallel surrogate-assisted global optimization with expensive functions—a survey. Struct Multidiscip Optim 54(1):3–13MathSciNetCrossRefGoogle Scholar
  37. 37.
    Schonlau M (1997) Computer experiments and global optimizationGoogle Scholar
  38. 38.
    Li Y, Wu Y, Zhao J, Chen L (2017) A Kriging-based constrained global optimization algorithm for expensive black-box functions with infeasible initial points. J Glob Optim 67(1–2):343–366MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wang Z, Ierapetritou M (2018) Constrained optimization of black-box stochastic systems using a novel feasibility enhanced Kriging-based method. Comput Chem Eng 118:210–230CrossRefGoogle Scholar
  40. 40.
    Zhang Y, Han ZH, Zhang KS (2018) Variable-fidelity expected improvement method for efficient global optimization of expensive functions. Struct Multidiscip Optim 58:1431–1451MathSciNetCrossRefGoogle Scholar
  41. 41.
    Parr JM, Keane AJ, Forrester AIJ, Holden CME (2012) Infill sampling criteria for surrogate-based optimization with constraint handling. Eng Optim 44(10):1147–1166CrossRefzbMATHGoogle Scholar
  42. 42.
    Sasena MJ, Papalambros P, Goovaerts P (2002) Exploration of metamodeling sampling criteria for constrained global optimization. Eng optim 34(3):263–278CrossRefGoogle Scholar
  43. 43.
    Bichon BJ, Eldred MS, Swiler LP, Mahadevan S, McFarland JM (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46(10):2459–2468CrossRefGoogle Scholar
  44. 44.
    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 162:102–115CrossRefGoogle Scholar
  45. 45.
    Shu L, Jiang P, Wan L, Zhou Q, Shao X, Zhang Y (2017) Metamodel-based design optimization employing a novel sequential sampling strategy. Eng Comput 34(8):2547–2564CrossRefGoogle Scholar
  46. 46.
    Liu H, Xu S, Chen X, Wang X, Ma Q (2016) Constrained global optimization via a DIRECT-type constraint-handling technique and an adaptive metamodeling strategy. Struct Multidiscip Optim 55:155–177MathSciNetCrossRefGoogle Scholar
  47. 47.
    Dong H, Song B, Dong Z, Wang P (2016) Multi-start space reduction (MSSR) surrogate-based global optimization method. Struct Multidiscip Optim 54(4):907–926CrossRefGoogle Scholar
  48. 48.
    Shi R, Liu L, Long T, Wu Y, Tang Y (2019) Filter-based adaptive Kriging method for black-box optimization problems with expensive objective and constraints. Comput Methods Appl Mech Eng 347:782–805MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wu Y, Yin Q, Jie H, Wang B, Zhao J (2018) A RBF-based constrained global optimization algorithm for problems with computationally expensive objective and constraints. Struct Multidiscip Optim pp. 1–23Google Scholar
  50. 50.
    Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4:409–423MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Zhou Q, Wang Y, Choi S-K, Jiang P, Shao X, Hu J (2017) A sequential multi-fidelity metamodeling approach for data regression. Knowl-Based Syst 134:199–212CrossRefGoogle Scholar
  52. 52.
    Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127CrossRefGoogle Scholar
  53. 53.
    Zhu J, Wang Y-J, Collette M (2013) A multi-objective variable-fidelity optimization method for genetic algorithms. Eng Optim 46(4):521–542MathSciNetCrossRefGoogle Scholar
  54. 54.
    Zhou H, Zhou Q, Liu C, Zhou T (2018) A kriging metamodel-assisted robust optimization method based on a reverse model. Eng Optim 50(2):253–272CrossRefGoogle Scholar
  55. 55.
    Wang Z, Ierapetritou M (2018) Constrained optimization of black-box stochastic systems using a novel feasibility enhanced Kriging-based method. Comput Chem Eng 118:210–223CrossRefGoogle Scholar
  56. 56.
    Garcia S, Herrera F (2008) An extension on “statistical comparisons of classifiers over multiple data sets” for all pairwise comparisons. J Mach Learn Res 9:2677–2694zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Jiachang Qian
    • 1
    • 2
  • Jiaxiang Yi
    • 1
  • Yuansheng Cheng
    • 1
    • 3
  • Jun Liu
    • 1
    • 3
  • Qi Zhou
    • 4
    Email author
  1. 1.School of Naval Architecture and Ocean EngineeringHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.Wuhan Second Ship Design and Research InstituteWuhanPeople’s Republic of China
  3. 3.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE)ShanghaiPeople’s Republic of China
  4. 4.School of Aerospace EngineeringHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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