Structural and Multidisciplinary Optimization

, Volume 59, Issue 2, pp 485–506 | Cite as

Multi-surrogate-based global optimization using a score-based infill criterion

  • Huachao DongEmail author
  • Siqing Sun
  • Baowei Song
  • Peng Wang
Research Paper


This paper presents a new global optimization algorithm named MGOSIC to solve unconstrained expensive black-box optimization problems. In MGOSIC, three surrogate models Kriging, Radial Basis Function (RBF), and Quadratic Response Surfaces (QRS) are dynamically constructed, respectively. Additionally, a multi-point infill criterion is proposed to obtain new points in each cycle, where a score-based strategy is presented to mark cheap points generated by Latin hypercube sampling. According to their predictive values from the three surrogate models, the promising cheap points are assigned with different scores. In order to obtain the samples with diversity, a Max-Min approach is proposed to select promising sample points from the cheap point sets with higher scores. Simultaneously, the best solutions predicted by Kriging, RBF, and QRS are also recorded as supplementary samples, respectively. Once MGOSIC gets stuck in a local valley, the estimated mean square error of Kriging will be maximized to explore the sparsely sampled regions. Moreover, the whole optimization algorithm is carried out alternately in the global space and a reduced space. In summary, MGOSIC not only brings a new idea for multi-point sampling, but also builds a reasonable balance between exploitation and exploration. Finally, 19 mathematical benchmark cases and an engineering application of hydrofoil optimization are used to test MGOSIC. Furthermore, seven existing global optimization algorithms are also tested as contrast. The final results show that MGOSIC has high efficiency, strong stability, and better multi-point sampling capability in dealing with expensive black-box optimization problems.


Kriging model Quadratic response surface Radial basis function Expensive black-box problems Multi-point infilling criterion 



This project is supported by National Natural Science Foundation of China (Grant No. 51805436). The authors are also grateful to members of the research group for the implementation of some existing global optimization algorithms and benchmark test cases.

Funding information

This project is supported by National Natural Science Foundation of China (Grant No. 51805436).


  1. Cai X, Qiu H, Gao L et al (2017) A multi-point sampling method based on kriging for global optimization. Struct Multidiscip Optim 56(1):71–88MathSciNetCrossRefGoogle Scholar
  2. Dong H, Song B, Wang P et al (2015a) Multi-fidelity information fusion based on prediction of kriging. Struct Multidiscip Optim 51(6):1267–1280CrossRefGoogle Scholar
  3. Dong H, Song B, Wang P et al (2015b) A kind of balance between exploitation and exploration on kriging for global optimization of expensive functions. J Mech Sci Technol 29(5):2121–2133CrossRefGoogle Scholar
  4. Dong H, Song B, Dong Z et al (2016) Multi-start space reduction (MSSR) surrogate-based global optimization method. Struct Multidiscip Optim 54(4):907–926CrossRefGoogle Scholar
  5. Dong H, Song B, Wang P et al (2018) Surrogate-based optimization with clustering-based space exploration for expensive multimodal problems. Struct Multidiscip Optim 57(4):1553–1577CrossRefGoogle Scholar
  6. Dyn N, Levin D, Rippa S (1986) Numerical procedures for surface fitting of scattered data by radial functions. SIAM J Sci Stat Comput 7(2):639–659MathSciNetCrossRefzbMATHGoogle Scholar
  7. Eglajs V, Audze P (1977) New approach to the design of multifactor experiments. Probl Dyn Strengths 35(1):104–107Google Scholar
  8. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1):50–79CrossRefGoogle Scholar
  9. Glaz B, Friedmann PP, Liu L (2008) Surrogate based optimization of helicopter rotor blades for vibration reduction in forward flight. Struct Multidiscip Optim 35(4):341–363CrossRefGoogle Scholar
  10. Gu J, Li G, Dong Z (2012) Hybrid and adaptive meta-model-based global optimization. Eng Optim 44(1):87–104CrossRefGoogle Scholar
  11. Gu J, Li G, Gan N (2017) Hybrid metamodel-based design space management method for expensive problems. Eng Optim 49(9):1573–1588MathSciNetCrossRefGoogle Scholar
  12. Gutmann HM (2001) A radial basis function method for global optimization. J Glob Optim 19(3):201–227MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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
  14. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915CrossRefGoogle Scholar
  15. Jamil M, Yang XS (2013) A literature survey of benchmark functions for global optimisation problems. Int J Math Model Numer Optim 4(2):150–194zbMATHGoogle Scholar
  16. Jiang F, Xia H, Tran QA et al (2017) A new binary hybrid particle swarm optimization with wavelet mutation. Knowl-Based Syst 130:90–101CrossRefGoogle Scholar
  17. Jin R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. J Stat Plan Inference 134(1):268–287MathSciNetCrossRefzbMATHGoogle Scholar
  18. Jones DR, Schonlau M, Welch WJ (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kleijnen JPC (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716MathSciNetCrossRefzbMATHGoogle Scholar
  20. Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. J South Afr Inst Min Metall 52(6):119–139Google Scholar
  21. Krityakierne T, Akhtar T, Shoemaker CA (2016) SOP: parallel surrogate global optimization with Pareto center selection for computationally expensive single objective problems. J Glob Optim 66(3):417–437MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kulfan BM (2008) Universal parametric geometry representation method. J Aircr 45(1):142–158CrossRefGoogle Scholar
  23. Lakshika E, Barlow M, Easton A (2017) Understanding the interplay of model complexity and fidelity in multiagent systems via an evolutionary framework. IEEE Trans Comput Intell AI in Games 9(3):277–289CrossRefGoogle Scholar
  24. Li Z, Ruan S, Gu J et al (2016) Investigation on parallel algorithms in efficient global optimization based on multiple points infill criterion and domain decomposition. Struct Multidiscip Optim 54(4):747–773MathSciNetCrossRefGoogle Scholar
  25. Long T, Wu D, Guo X et al (2015) Efficient adaptive response surface method using intelligent space exploration strategy. Struct Multidiscip Optim 51(6):1335–1362CrossRefGoogle Scholar
  26. Masters DA, Taylor NJ, Rendall TCS et al (2017) Multilevel subdivision parameterization scheme for aerodynamic shape optimization. AIAA J 55:3288–3303CrossRefGoogle Scholar
  27. Meng Z, Pan JS, Xu H (2016) QUasi-affine TRansformation evolutionary (QUATRE) algorithm: a cooperative swarm based algorithm for global optimization. Knowl-Based Syst 109:104–121CrossRefGoogle Scholar
  28. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61CrossRefGoogle Scholar
  29. Müller J (2012) User guide for modularized surrogate model toolbox. Department of Mathematics, Tampere University of technology, TampereGoogle Scholar
  30. Myers RH, Montgomery DC, Vining GG et al (2004) Response surface methodology: a retrospective and literature survey. J Qual Technol 36(1):53CrossRefGoogle Scholar
  31. Nocedal J, Wright S (2006) Numerical Optimization. Springer, BerlinzbMATHGoogle Scholar
  32. Ong YS, Nair PB, Keane AJ (2003) Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J 41(4):687–696CrossRefGoogle Scholar
  33. Pan WT (2012) A new fruit fly optimization algorithm: taking the financial distress model as an example. Knowl-Based Syst 26:69–74CrossRefGoogle Scholar
  34. Queipo NV, Haftka RT, Shyy W et al (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28CrossRefGoogle Scholar
  35. Regis RG, Shoemaker CA (2007a) A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J Comput 19(4):497–509MathSciNetCrossRefzbMATHGoogle Scholar
  36. Regis RG, Shoemaker CA (2007b) Improved strategies for radial basis function methods for global optimization. J Glob Optim 37(1):113–135MathSciNetCrossRefzbMATHGoogle Scholar
  37. Regis RG, Shoemaker CA (2013) A quasi-multistart framework for global optimization of expensive functions using response surface models. J Glob Optim 56(4):1719–1753MathSciNetCrossRefzbMATHGoogle Scholar
  38. Rocca P, Oliveri G, Massa A (2011) Differential evolution as applied to electromagnetics. IEEE Antennas Propag Mag 53(1):38–49CrossRefGoogle Scholar
  39. Sacks J, Welch WJ, Mitchell TJ et al (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–435MathSciNetCrossRefzbMATHGoogle Scholar
  40. Sala R, Baldanzini N, Pierini M (2016) Representative surrogate problems as test functions for expensive simulators in multidisciplinary design optimization of vehicle structures. Struct Multidiscip Optim 54(3):449–468MathSciNetCrossRefGoogle Scholar
  41. Singh P, Van Der Herten J, Deschrijver D et al (2017) A sequential sampling strategy for adaptive classification of computationally expensive data. Struct Multidiscip Optim 55(4):1425–1438MathSciNetCrossRefGoogle Scholar
  42. Sun C, Zeng J, Pan J et al (2013) A new fitness estimation strategy for particle swarm optimization. Inf Sci 221(2):355–370MathSciNetCrossRefzbMATHGoogle Scholar
  43. Tyan M, Nguyen NV, Lee JW (2015) Improving variable-fidelity modelling by exploring global design space and radial basis function networks for aerofoil design. Eng Optim 47(7):885–908CrossRefGoogle Scholar
  44. Viana FAC, Haftka RT, Watson LT (2013) Efficient global optimization algorithm assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689CrossRefzbMATHGoogle Scholar
  45. Wang L, Shan S, Wang GG (2004) Mode-pursuing sampling method for global optimization on expensive black-box functions. Eng Optim 36(4):419–438CrossRefGoogle Scholar
  46. Wang H, Fan T, Li G (2017a) Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems. Struct Multidiscip Optim 55(6):2143–2157CrossRefGoogle Scholar
  47. Wang L, Pei J, Menhas MI et al (2017b) A hybrid-coded human learning optimization for mixed-variable optimization problems. Knowl-Based Syst 127:114–125CrossRefGoogle Scholar
  48. Yang XS (2010) A new metaheuristic bat-inspired algorithm. Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), pp 65–74Google Scholar
  49. Younis A, Dong Z (2010) Metamodelling and search using space exploration and unimodal region elimination for design optimization. Eng Optim 42(6):517–533CrossRefGoogle Scholar
  50. Zhou G, Zhao W, Li Q et al (2017) Multi-objective robust design optimization of a novel NPR energy absorption structure for vehicles front ends to enhance pedestrian lower leg protection. Struct Multidiscip Optim 56(5):1215–1224CrossRefGoogle Scholar
  51. Zhou Q, Wang Y, Choi SK et al (2017a) A sequential multi-fidelity metamodeling approach for data regression. Knowl-Based Syst 134:199–212CrossRefGoogle Scholar
  52. Zhou Q, Jiang P, Shao X et al (2017b) A variable fidelity information fusion method based on radial basis function. Adv Eng Inform 32:26–39CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Marine Science and TechnologyNorthwestern Polytechnical UniversityXi’anChina
  2. 2.Department of Mechanical EngineeringUniversity of VictoriaVictoriaCanada

Personalised recommendations