Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Parallelized multiobjective efficient global optimization algorithm and its applications


In engineering practice, most optimization problems have multiple objectives, which are usually in a form of expensive black-box functions. The multiobjective efficient global optimization (MOEGO) algorithms have been proposed recently to sequentially sample the design space, aiming to seek for optima with a minimum number of sampling points. With the advance in computing resources, it is wise to make optimization parallelizable to shorten the total design cycle further. In this study, two different parallelized multiobjective efficient global optimization algorithms were proposed on the basis of the Kriging modeling technique. With use of the multiobjective expectation improvement, the proposed algorithm is able to balance local exploitation and global exploration. To implement parallel computing, the “Kriging Believer” and “multiple good local optima” strategies were adopted here to develop new sample infill criteria for multiobjective optimization problems. The proposed algorithms were applied to five mathematical benchmark examples first, which demonstrated faster convergence and better accuracy with more uniform distribution of Pareto points, in comparison with the two other conventional algorithms. The best performed “Kriging Believer” strategy approach was then applied to two more sophisticated real-life engineering case studies on the tailor-rolled blank (TRB) structures for crashworthiness design. After optimization, the TRB hat-shaped tube achieved a 3% increase in energy absorption and a 10.7% reduction in mass, and the TRB B-pillar attained a 10.1% reduction in mass and a 12.8% decrease in intrusion, simultaneously. These benchmark and engineering examples demonstrated that the proposed methods are fairly promising for being an effective tool for a range of design problems.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26


  1. Akhtar T, Shoemaker CA (2016) Multiobjective optimization of computationally expensive multi-modal functions with RBF surrogates and multi-rule selection. J Glob Optim 64(1):17–32

  2. Chankong V, Haimes YY (2008) Multiobjective decision making: theory and methodology. Courier Dover Publications

  3. Chen G, Han X, Liu G, Jiang C, Zhao Z (2012) An efficient multi-objective optimization method for black-box functions using sequential approximate technique. Appl Soft Comput 12(1):14–27

  4. Chevalier C, Ginsbourger D (2013, January) Fast computation of the multi-points expected improvement with applications in batch selection. In: International Conference on Learning and Intelligent Optimization. Springer, Berlin, Heidelberg, pp 59–69

  5. Chokshi P, Dashwood R, Hughes DJ (2017) Artificial Neural Network (ANN) based microstructural prediction model for 22MnB5 boron steel during tailored hot stamping. Comput Struct 190:162–172

  6. Coello CC, Lechuga MS (2002) MOPSO: a proposal for multiple objective particle swarm optimization. In: Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600), vol 2. IEEE, pp 1051–1056

  7. Coello CA, Pulido GT (2001) Multiobjective optimization using a micro-genetic algorithm. In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation. Morgan Kaufmann Publishers Inc, pp 274–282

  8. Cressie NA (1993) In: NAC C (ed) Spatial prediction and kriging. Statistics for Spatial Data. John Wiley & Sons, New York, pp 105–209

  9. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

  10. Deb K, Mohan M, Mishra S (2003, April) Towards a quick computation of well-spread pareto-optimal solutions. In: International Conference on Evolutionary Multi-Criterion Optimization. Springer, Berlin, Heidelberg, pp 222–236

  11. Dixon LCW, Szego GP (1978) The optimization problem: an introduction. Towards Global Optimization II. North Holland, New York

  12. Doerner K, Gutjahr WJ, Hartl RF, Strauss C, Stummer C (2004) Pareto Ant colony optimization: a metaheuristic approach to multiobjective portfolio selection. Ann Oper Res 131(1-4):79–99

  13. Duan L, Xiao NC, Li G, Xu F, Chen T, Cheng A (2017) Bending analysis and design optimisation of tailor-rolled blank thin-walled structures with top-hat sections. Int J Crashworthiness 22(3):227–242

  14. Emmerich MTM, Giannakoglou KC, Naujoks B (2006) Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans Evol Comput 10(4):421–439

  15. Fang H, Rais-Rohani M, Liu Z, Horstemeyer MF (2005) A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput Struct 83(25-26):2121–2136

  16. Fang J, Gao Y, Sun G, Qiu N, Li Q (2015a) On design of multi-cell tubes under axial and oblique impact loads. Thin-Walled Struct 95:115–126

  17. Fang J, Gao Y, Sun G, Zheng G, Li Q (2015b) Dynamic crashing behavior of new extrudable multi-cell tubes with a functionally graded thickness. Int J Mech Sci 103:63–73

  18. Fang J, Gao Y, Sun G, Xu C, Li Q (2016) Multiobjective sequential optimization for a vehicle door using hybrid materials tailor-welded structure. Proc Inst Mech Eng C J Mech Eng Sci 230(17):3092–3100

  19. Fang J, Sun G, Qiu N, Kim NH, Li Q (2017) On design optimization for structural crashworthiness and its state of the art. Struct Multidiscip Optim 55(3):1091–1119

  20. Farina M (2002) A neural network based generalized response surface multiobjective evolutionary algorithm. In: Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600), vol 1. IEEE, pp 956–961

  21. Fonseca, C. M., & Fleming, P. J. (1995). Multiobjective optimization and multiple constraint handling with evolutionary algorithms 1: a unified formulation

  22. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79

  23. Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. John Wiley & Sons

  24. Ginsbourger D, Riche RL, Carraro L (2008) A multi-points criterion for deterministic parallel global optimization based on Gaussian processes. J Glob Optim Revision

  25. Gutmann HM (2001) A radial basis function method for global optimization. J Glob Optim 19(3):201–227

  26. Haftka RT, Villanueva D, Chaudhuri A (2016) Parallel surrogate-assisted global optimization with expensive functions–a survey. Struct Multidiscip Optim 54(1):3–13

  27. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915

  28. Hernández-Lobato JM, Hoffman MW, Ghahramani Z (2014) Predictive entropy search for efficient global optimization of black-box functions. In: Advances in neural information processing systems, pp 918–926

  29. Hou S, Han X, Sun G, Long S, Li W, Yang X, Li Q (2011) Multiobjective optimization for tapered circular tubes. Thin-Walled Struct 49(7):855–863

  30. Jeong S, Obayashi S (2005) Efficient global optimization (EGO) for multi-objective problem and data mining. In: 2005 IEEE congress on evolutionary computation, vol 3. IEEE, pp 2138–2145

  31. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

  32. Karakasis MK, Giannakoglou KC (2006) On the use of metamodel-assisted, multi-objective evolutionary algorithms. Eng Optim 38(8):941–957

  33. Keane AJ (2006) Statistical improvement criteria for use in multiobjective design optimization. AIAA J 44(4):879–891

  34. Knowles J, Hughes EJ (2005) Multiobjective optimization on a budget of 250 evaluations. In: International Conference on Evolutionary Multi-criterion Optimization. Springer, Berlin, Heidelberg, pp 176–190

  35. Luo C, Shimoyama K, Obayashi S (2014) Kriging model based many-objective optimization with efficient calculation of expected hypervolume improvement. In: 2014 IEEE Congress on Evolutionary Computation (CEC), pp. 1187-1194. Beijing, China

  36. Martínez SZ, Coello CAC (2013) Combining surrogate models and local search for dealing with expensive multi-objective optimization problems. In: 2013 IEEE Congress on Evolutionary Computation, pp 2572–2579

  37. McKay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

  38. Pan F, Zhu P, Zhang Y (2010) Metamodel-based lightweight design of B-pillar with TWB structure via support vector regression. Comput Struct 88(1–2):36–44

  39. Poloni, C. (1995). Hybrid GA for multi objective aerodynamic shape optimization.

  40. Ponweiser W, Wagner T, Biermann D, Vincze M (2008) Multiobjective optimization on a limited budget of evaluations using model-assisted $\mathcal {S} $-metric selection. In: International Conference on Parallel Problem Solving from Nature. Springer, Berlin, Heidelberg, pp 784–794

  41. Qiu N, Gao Y, Fang J, Feng Z, Sun G, Li Q (2015) Crashworthiness analysis and design of multi-cell hexagonal columns under multiple loading cases. Finite Elem Anal Des 104:89–101

  42. Regis RG, Shoemaker CA (2005) Constrained global optimization of expensive black box functions using radial basis functions. J Glob Optim 31(1):153–171

  43. Shimoyama K, Yoshimizu S, Jeong S, Obayashi S, Yokono Y (2011) Multi-objective design optimization for a steam turbine stator blade using LES and GA. J Comput Sci Technol 5(3):134–147

  44. Sóbester A, Leary SJ, Keane AJ (2004) A parallel updating scheme for approximating and optimizing high fidelity computer simulations. Struct Multidiscip Optim 27(5):371–383

  45. Sun G, Li G, Hou S, Zhou S, Li W, Li Q (2010) Crashworthiness design for functionally graded foam-filled thin-walled structures. Mater Sci Eng A 527(7–8):1911–1919

  46. Sun G, Xu F, Li G, Li Q (2014) Crashing analysis and multiobjective optimization for thin-walled structures with functionally graded thickness. Int J Impact Eng 64(64):62–74

  47. Sun G, Zhang H, Lu G, Guo J, Cui J, Li Q (2016) An experimental and numerical study on quasi-static and dynamic crashworthiness for tailor rolled blank (TRB) structures. Mater Des 118

  48. Sun G, Pang T, Fang J, Li G, Li Q (2017a) Parameterization of criss-cross configurations for multiobjective crashworthiness optimization. Int J Mech Sci 124:145–157

  49. Sun G, Zhang H, Fang J, Li G, Li Q (2017b) Multi-objective and multi-case reliability-based design optimization for tailor rolled blank (TRB) structures. Struct Multidiscip Optim 55(5):1899–1916

  50. Van DA, Gary V, Lamont B (1999) Multiobjective evolutionary algorithm research: a history and analysis. Evol Comput 8(2):125–147

  51. Vincenzi L, Gambarelli P (2017) A proper infill sampling strategy for improving the speed performance of a surrogate-assisted evolutionary algorithm. Comput Struct 178:58–70

  52. Voß T, Hansen N, Igel C (2010) Improved step size adaptation for the MO-CMA-ES. In: Proceedings of the 12th annual conference on Genetic and evolutionary computation. ACM, pp 487–494

  53. Wang D, Wu Z, Fei Y, Zhang W (2014) Structural design employing a sequential approximation optimization approach. Comput Struct 134(4):75–87

  54. Xu F, Sun G, Li G, Li Q (2013) Crashworthiness design of multi-component tailor-welded blank (TWB) structures. Struct Multidiscip Optim 48(3):653–667

  55. Yang Z, Peng Q, Yang J (2012) Lightweight design of B-pillar with TRB concept considering crashworthiness. In: 2012 Third International Conference on Digital Manufacturing & Automation. IEEE, pp 510–513

  56. Yun Y, Min Y, Nakayama H (2009) Multi-objective optimization based on meta-modeling by using support vector regression. Optim Eng 10(2):167–181

  57. Zhang Y, Sun G, Li G, Luo Z, Li Q (2012) Optimization of foam-filled bitubal structures for crashworthiness criteria. Mater Des 38:99–109

  58. Zitzler, E., & Thiele, L. (1998a). An evolutionary algorithm for multiobjective optimization: the strength pareto approach. TIK-report, 43

  59. Zitzler E, Thiele L (1998b) Multiobjective optimization using evolutionary algorithms—a comparative case study. In: International conference on parallel problem solving from nature. Springer, Berlin, Heidelberg, pp 292–301

  60. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195

  61. Zitzler, E., Laumanns, M., & Thiele, L. (2001). SPEA2: Improving the strength Pareto evolutionary algorithm. TIK-report, 103

  62. Zuluaga M, Krause A, Püschel M (2016) ε-pal: an active learning approach to the multi-objective optimization problem. J Mach Learn Res 17(1):3619–3650

  63. Zurada JM (1992) Introduction to artificial neural systems, vol 8. West publishing company, St. Paul

Download references


This work is supported by National Natural Science Foundation of China (51575172) and Australian Research Council (ARC) (DP190103752). Dr Guangyong Sun is a recipient of ARC Discovery Early Career Research Award (DECRA).

Author information

Correspondence to Guangyong Sun.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Responsible Editor: Nestor V Queipo

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, G., Tian, Y., Wang, R. et al. Parallelized multiobjective efficient global optimization algorithm and its applications. Struct Multidisc Optim 61, 763–786 (2020).

Download citation


  • Parallel computing
  • Efficient global optimization (EGO)
  • Multiobjective optimization
  • Kriging Believer
  • Multiple good local optima