Structural and Multidisciplinary Optimization

, Volume 58, Issue 4, pp 1633–1655 | Cite as

A RBF-based constrained global optimization algorithm for problems with computationally expensive objective and constraints

  • Yizhong Wu
  • Qian Yin
  • Haoxiang Jie
  • Boxing Wang
  • Jianjun Zhao


This paper presents a metamodel-based constrained optimization method, called Radial basis function-based Constrained Global Optimization (RCGO), to solve optimization problems involving computationally expensive objective function and inequality constraints. RCGO is an extension of the adaptive metamodel-based global optimization (AMGO) algorithm which can handle unconstrained black-box optimization problems. Firstly, a sequential sampling method is implemented to obtain the initial points for building the radial basis function (RBF) approximations to all computational expensive functions while enforcing a feasible solution. Then, an auxiliary objective function subject to the approximate constraints is constructed to determine the next iterative point and further improve the solution. During the process, a distance function with a group of exponents is introduced in the auxiliary function to balance the local exploitation and the global exploration. The RCGO method is tested on a series of benchmark problems, and the results demonstrate that RCGO needs fewer costly evaluations and can be applied for costly constrained problems with all infeasible start points. And the test results on the 30D problems demonstrate that RCGO has advantages in solving the problems. The proposed method is then applied to the design of a cycloid gear pump and desirable results are obtained.


Metamodel Radial basis function Constrained optimization Computationally expensive function Global optimization 



This work is supported by the National Natural Science Foundation of China (grant numbers 51575205 and 61672247).


  1. Adams BM, Bauman LE, Bohnhoff WJ et al (2013) Dakota, a multilevel parallel objectoriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 5.3.1 user’s manual. Technical Report SAND2010-2183. Sandia National Laboratories, Albuquerque. Updated Jan. 2013Google Scholar
  2. Alexandrov NM, Dennis JE, Lewis RM et al (1998) A trust-region framework for managing the use of approximation models in optimization. Structural Optimization 15(1):16–23CrossRefGoogle Scholar
  3. Amouzgar K, Stromberg N (2017) Radial basis functions as surrogate models with a priori bias in comparison with a posteriori bias. Struct Multidiscip Optim 55(4):1453–1469MathSciNetCrossRefGoogle Scholar
  4. Arora J (2004) Introduction to optimum design. Elsevier Academic Press, LondonGoogle Scholar
  5. Basudhar A, Dribusch C, Lacaze S et al (2012) Constrained efficient global optimization with support vector machines. Struct Multidiscip Optim 46(2):201–221CrossRefzbMATHGoogle Scholar
  6. Bouhlel MA, Bartoli N, Otsmane A et al (2016) Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction. Struct Multidiscip Optim 53(5):935–952MathSciNetCrossRefGoogle Scholar
  7. Box GE, Draper NR (1987) Empirical model-building and response surfaces. WileyGoogle Scholar
  8. Buhmann MD (2000) Radial basis functions. Acta Numer 9(1):1–38MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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
  10. Cassioli A, Schoen F (2011) Global optimization of expensive black box problems with a known lower bound. J Glob Optim:1–14Google Scholar
  11. Chaudhuri A, Haftka RT (2014) Efficient Global Optimization with Adaptive Target Setting. AIAA J 52(7):1573–1578CrossRefGoogle Scholar
  12. Coello CAC, Montes EM (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16(3):193–203CrossRefGoogle Scholar
  13. Competition & Special Session on Constrained Real-Parameter Optimization (2010) Available from:
  14. Ferreira WG, Serpa AL (2018) Ensemble of metamodels: extensions of the least squares approach to efficient global optimization. Struct Multidiscip Optim 57(1):131–159Google Scholar
  15. Floudas CA, Pardalos PM (1990) A collection of test problems for constrained global optimization algorithms. Springer, BerlinCrossRefzbMATHGoogle Scholar
  16. Franke R (1982) Scattered data interpolation: tests of some methods. Math Comput 38(157):181–200MathSciNetzbMATHGoogle Scholar
  17. Friedman JH (1991) Multivariate adaptive regression splines. The annals of statistics:1–67Google Scholar
  18. Friedman JH (1993) Fast MARS. Stanford University. Dept. of Statistics. Laboratory for Computational StatisticsGoogle Scholar
  19. Gano SE, Renaud JE, Martin JD et al (2006) Update strategies for kriging models used in variable fidelity optimization. Struct Multidiscip Optim 32(4):287–298CrossRefGoogle Scholar
  20. Ginsbourger D, Le Riche R, Carraro L (2010) Kriging is well-suited to parallelize optimization. In: Tenne Y, Goh C-K (eds) Computational intelligence in expensive optimization problems. Springer Berlin Heidelberg, Berlin, pp 131–162CrossRefGoogle Scholar
  21. Goel T, Haftka RT, Shyy W et al (2007) Ensemble of surrogates. Struct Multidiscip Optim 33(3):199–216CrossRefGoogle Scholar
  22. Gu J, Li GY, Dong Z (2012) Hybrid and adaptive meta-model-based global optimization. Eng Optim 44(1):87–104CrossRefGoogle Scholar
  23. Gunn SR (1998) Support vector machines for classification and regression. ISIS technical report, vol 14Google Scholar
  24. Gutmann H-M (2001) A radial basis function method for global optimization. J Glob Optim 19(3):201–227MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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
  26. Hesse R (1973) A heuristic search procedure for estimating a global solution of nonconvex programming problems. Oper Res 21(6):1267–1280MathSciNetCrossRefzbMATHGoogle Scholar
  27. Huang D, Allen TT, Notz WI et al (2006a) Global optimization of stochastic black-box systems via sequential kriging meta-models. J Glob Optim 34(3):441–466MathSciNetCrossRefzbMATHGoogle Scholar
  28. Huang D, Allen TT, Notz WI et al (2006b) Sequential kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 32(5):369–382CrossRefGoogle Scholar
  29. Jie H, Wu Y, Ding J (2015) An adaptive metamodel-based global optimization algorithm for black-box type problems. Eng Optim 47(11):1459–1480MathSciNetCrossRefGoogle Scholar
  30. Jones DR (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Opt 21(4):345–383MathSciNetCrossRefzbMATHGoogle Scholar
  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. Kazemi M, Wang GG, Rahnamayan S et al (2011) Metamodel-Based Optimization for Problems With Expensive Objective and Constraint Functions. Journal of Mechanical Design 133(1):014505-014505-7CrossRefGoogle Scholar
  33. Kerry K, Hawick KA (1998) Kriging interpolation on high-performance computers. In: High-Performance computing and networking. Springer Berlin Heidelberg, New York, pp 429–438CrossRefGoogle Scholar
  34. Kitayama S, Arakawa M, Yamazaki K (2011) Sequential approximate optimization using radial basis function network for engineering optimization. Optim Eng 12(4):535–557CrossRefzbMATHGoogle Scholar
  35. Krishnamurthy T (2003) Response surface approximation with augmented and compactly supported radial basis functions. In: Proceeding of the 44 th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. NorfolkGoogle Scholar
  36. Lee HKH, Gramacy RB, Linkletter C et al (2011) Optimization subject to hidden constraints vis statistical emulation. Pacific Journal of Optimization 7(3):467–478MathSciNetzbMATHGoogle Scholar
  37. 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
  38. Long T, Wu D, Guo X et al (2015) Efficient adaptive response surface method using intelligent space exploration strategy. Struct Multidisc Optim 51(6):1335–1362CrossRefGoogle Scholar
  39. Lowe D, Broomhead D (1988) Multivariable functional interpolation and adaptive networks. Complex Syst 2:321–355MathSciNetzbMATHGoogle Scholar
  40. Mallipeddi R, Suganthan PN (2010) Problem definitions and evaluation criteria for the CEC 2010 competition on constrained real-parameter optimization. Nanyang Technological UniversityGoogle Scholar
  41. McDonald DB, Grantham WJ, Tabor WL et al (2007) Global and local optimization using radial basis function response surface models. Appl Math Model 31(10):2095–2110CrossRefzbMATHGoogle Scholar
  42. Mezura-Montes E, Cetina-Domínguez O (2012) Empirical analysis of a modified artificial bee colony for constrained numerical optimization. Appl Math Comput 218(22):10943–10973MathSciNetzbMATHGoogle Scholar
  43. Michalewicz Z, Schoenauer M (1996) Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 4(1):1–32CrossRefGoogle Scholar
  44. Myers RH, Anderson-Cook CM (2009) Response surface methodology: process and product optimization using designed experiments, vol 705. WileyGoogle Scholar
  45. Nakayama H, Arakawa M, Washino K (2003) Optimization for black-box objective functions. Optimization and optimal control, pp 185–210Google Scholar
  46. Picheny V, Wagner T, Ginsbourger D (2013) A benchmark of kriging-based infill criteria for noisy optimization. Struct Multidiscip Optim 48(3):607–626CrossRefGoogle Scholar
  47. Powell MJ (1992) The theory of radial basis function approximation in 1990. Adv Numer Anal 2:105–210MathSciNetzbMATHGoogle Scholar
  48. Powell MJ (1999) Recent research at Cambridge on radial basis functions. SpringerGoogle Scholar
  49. Regis RG (2011) Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions. Comput Oper Res 38(5):837–853MathSciNetCrossRefGoogle Scholar
  50. Regis RG (2014a) Constrained optimization by radial basis function interpolation for high-dimensional expensive black-box problems with infeasible initial points. Engineering Optimization 46(2):218–243MathSciNetCrossRefGoogle Scholar
  51. Regis RG (2014b) Evolutionary programming for high-dimensional constrained expensive black-box optimization using radial basis functions. IEEE Trans Evol Comput 18(3):326–347CrossRefGoogle Scholar
  52. Regis RG, Shoemaker CA (2005) Constrained global optimization of expensive black box functions using radial basis functions. J Glob Optim 31(1):153–171MathSciNetCrossRefzbMATHGoogle Scholar
  53. Regis RG, Shoemaker CA (2007a) A stochastic radial basis function method for the global optimization of expensive functions. Informs Journal on Computing 19(4):497–509MathSciNetCrossRefzbMATHGoogle Scholar
  54. Regis RG, Shoemaker CA (2007b) Improved strategies for radial basis function methods for global optimization. J Glob Optim 37(1):113–135MathSciNetCrossRefzbMATHGoogle Scholar
  55. Regis RG, Wild SM (2017) CONORBIT: constrained optimization by radial basis function interpolation in trust regions. Optimization Methods & Software 32(3):552–580MathSciNetCrossRefzbMATHGoogle Scholar
  56. Sacks J, Welch WJ, Mitchell TJ et al (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–423MathSciNetCrossRefzbMATHGoogle Scholar
  57. Sasena MJ (2002) Flexibility and efficiency enhancements for constrainted global design optimization with kriging approximations. Ann Arbor, University of MichiganGoogle Scholar
  58. Sasena MJ, Papalambros P, Goovaerts P (2002) Exploration of metamodeling sampling criteria for constrained global optimization. Eng Opt 34(3):263–278CrossRefGoogle Scholar
  59. Schonlau M (1997) Computer experiments and global optimization. Waterloo, University of WaterlooGoogle Scholar
  60. Sóbester A, Leary SJ, Keane AJ (2004) A parallel updating scheme for approximating and optimizing high fidelity computer simulations. Struct Multidiscip Optim 27(5):371–383CrossRefGoogle Scholar
  61. Tang Y, Chen J, Wei J (2013) A surrogate-based particle swarm optimization algorithm for solving optimization problems with expensive black box functions. Engineering Optimization 45(5):557–576MathSciNetCrossRefGoogle Scholar
  62. Vapnik V, Golowich SE, Smola A (1997) Support vector method for function approximation, regression estimation, and signal processing. Adv Neural Inf Process Syst:281–287Google Scholar
  63. Viana F, Haftka R (2010) Surrogate-based optimization with parallel simulations using the probability of improvement. In: 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference. American Institute of Aeronautics and AstronauticsGoogle Scholar
  64. Viana FAC, Haftka RT, Watson LT (2013) Efficient global optimization algorithm assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689CrossRefzbMATHGoogle Scholar
  65. Wang GG, Shan S (2007) Review of metamodeling techniques in support of engineering design optimization. J Mech Des 129:370–380CrossRefGoogle Scholar
  66. Wang GG, Dong Z, Aitchison P (2001) Adaptive response surface method--a global optimization scheme for approximation-based design problems. Eng Optim 33(6):707–734CrossRefGoogle Scholar
  67. Wang LQ, Shan SQ, Wang GG (2004) Mode-pursuing sampling method for global optimization on expensive black-box functions. Eng Optim 36(4):419–438CrossRefGoogle Scholar
  68. Wei X (2012) Research of global optimization algorithm based on metamodel. Huazhong University of Science&TechnologyGoogle Scholar
  69. Wei X, Wu Y-Z, Chen L-P (2012) A new sequential optimal sampling method for radial basis functions. Appl Math Comput 218(19):9635–9646MathSciNetzbMATHGoogle Scholar
  70. Wild SM, Regis RG, Shoemaker CA (2008) ORBIT: optimization by radial basis function interpolation in trust-regions. SIAM J Sci Comput 30(6):3197–3219MathSciNetCrossRefzbMATHGoogle Scholar
  71. Wu Z (1997) Compactly supported radial functions and the Strang-Fix condition. Appl Math Comput 84(2):115–124MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Yizhong Wu
    • 1
  • Qian Yin
    • 1
  • Haoxiang Jie
    • 2
    • 3
  • Boxing Wang
    • 1
  • Jianjun Zhao
    • 1
  1. 1.National CAD Supported Software Engineering CentreHuazhong University of Science and TechnologyWuhanPeople’s Republic of China
  2. 2.Shanghai Marine Diesel Engine Research InstituteShanghaiPeople’s Republic of China
  3. 3.R&D Center, Micropowers Ltd.ShanghaiPeople’s Republic of China

Personalised recommendations