Journal of Global Optimization

, Volume 70, Issue 4, pp 757–781 | Cite as

An adaptive framework for costly black-box global optimization based on radial basis function interpolation

  • Zhe Zhou
  • Fusheng Bai


In this paper, we present a framework for the global optimization of costly black-box functions using response surface (RS) models. The main iteration steps of the framework which is referred to as the Adaptive Framework using Response Surface (ADFRS) consist of two phases. In the first phase, we implement a mixture of local searches and global searches to get a rough solution before the number of consecutive unsuccessful iterations exceeds a user-defined threshold. A procedure is embedded into this phase to check whether a small neighborhood of a global minimizer of the current RS model is fully explored or not, and then determine the search type (global search or local search) to be implemented next. Before performing a local search or a global search, the distance between the two global minimizers of the last and the current response surface models is checked, and the current global minimizer will be taken as the new evaluation point if this distance is very small. This strategy can quickly return a good evaluation point. In the second phase, we perform pure local search in the vicinity of the current best point to search for a better solution. Local searches are only implemented in the vicinities of the global minima of the RBF models in our scheme. Numerical experiments on some test problems are conducted to show the effectiveness of the present algorithm.


Global optimization Costly black-box functions Response surface model Radial basis function interpolation Local search Global search 



We would like to thank the two anonymous referees for their very helpful comments and insightful suggestions that have helped improve the presentation of this paper greatly.


  1. 1.
    Akhtar, T., Shoemaker, C.A.: Multi objective optimization of computationally costly multi-modal functions with RBF surrogates and multi-rule selection. J. Glob. Optim. 64, 17–32 (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Björkman, M., Holmström, K.: Global optimization of costly nonconvex functions using radial basis functions. Optim. Eng. 1(4), 373–397 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Box, G.E.P., Draper, N.R.: Empirical Model-Building and Response Surfaces. Wiley, New York (1987)zbMATHGoogle Scholar
  4. 5.
    Conn, A.R., Scheinberg, K., Toint, PhL: Recent progress in unconstrained nonlinear optimization without derivatives. Math. Program. 79(3), 397–414 (1997)MathSciNetzbMATHGoogle Scholar
  5. 7.
    Dixon, L.C.W., Szegö, G.: The global optimization problem: an introduction. In: Dixon, L.C.W., Szegö, G. (eds.) Towards Global Optimization 2, pp. 1–15. North-Holland, Amsterdam (1978)Google Scholar
  6. 8.
    Emmerich, M., Giotis, A., Özdemir, M., Bäck, T., Giannakoglou, K.: Metamodel-assisted evolution strategies. In: Parallel Problem Solving from Nature VII. Springer, pp. 361–370 (2002)Google Scholar
  7. 9.
    Friedman, J.H., Bentely, J., Finkel, R.A.: An algorithm for finding best matches in logarithmic expected time. ACM Trans. Math. Softw. 3, 209–226 (1977)CrossRefzbMATHGoogle Scholar
  8. 10.
    Gutmann, H.-M.: A radial basis function method for global optimization. J. Glob. Optim. 19(3), 201–227 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 12.
    Huang, D., Allen, T.T., Notz, W.I., Zeng, N.: Global optimization of stochastic black-box systems via sequential kriging meta-models. J. Glob. Optim. 34(3), 441–466 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 13.
    Jones, D.R.: Global optimization with response surfaces. In: Presented at the Fifth SIAM Conference on Optimization, Victoria, Canada (1996)Google Scholar
  11. 14.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 15.
    Khuri, A.I., Cornell, J.A.: Response Surfaces. Marcel Dekker Inc, New York (1987)zbMATHGoogle Scholar
  13. 16.
    Myers, R.H., Montgomery, D.C.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley, New York (1995)zbMATHGoogle Scholar
  14. 18.
    Powell, M.J.D.: The theory of radial basis function approximation in 1990. In: Light, W. (ed.) Advances in Numerical Analysis, Volume 2: Wavelets, Subdivision Algorithms and Radial Basis Functions, pp. 105–210. Oxford University Press, Oxford (1992)Google Scholar
  15. 19.
    Powell, M.J.D.: A direct search optimization method that models the objective and constraint functions by linear interpolation. In: Gomez, S., Hennart, J.-P. (eds.) Advances in Optimization and Numerical Analysis, pp. 51–67. Kluwer, Dordrecht (1994)CrossRefGoogle Scholar
  16. 21.
    Powell, M.J.D.: UOBYQA: unconstrained optimization by quadratic approximation. Math. Program. 92, 555–582 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 22.
    Powell, M.J.D.: On trust region methods for unconstrained minimization without derivatives, Technical Report. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK (2002)Google Scholar
  18. 23.
    Regis, R.G., Shoemaker, C.A.: Constrained global optimization of expensive black box functions using radial basis functions. J. Glob. Optim. 31, 153–171 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 24.
    Regis, R.G., Shoemaker, C.A.: Improved strategies for radial basis function methods for global optimization. J. Glob. Optim. 37(1), 113–135 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 25.
    Regis, R., Shoemaker, C.: A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J. Comput. 19, 497–509 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 26.
    Regis, R.G., Shoemaker, C.A.: Parallel stochastic global optimization using radial basis functions. INFORMS J. Comput. 21(3), 411–426 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 27.
    Regis, R.G., Shoemaker, C.A.: A quasi-multistart framework for global optimization of expensive functions using response surface methods. J. Glob. Optim. 56, 1719–1753 (2013)CrossRefzbMATHGoogle Scholar
  23. 28.
    Regis, R.G., Shoemaker, C.A.: Combining radial basis function surrogates dynamic coordinate search in high dimensional expensive black-box optimization. Eng. Optim. 45(5), 529–555 (2013)MathSciNetCrossRefGoogle Scholar
  24. 29.
    Schoen, F.: A wide class of test functions for global optimization. J. Glob. Optim. 3, 133–137 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 31.
    Törn, A., Zilinskas, A.: Glob. Optim. Springer, Berlin (1989)Google Scholar
  26. 32.
    Wild, S.M., Regis, R.G., Shoemaker, C.A.: ORBIT: optimization by radial basis function interpolation in trust-regions. SIAM J. Sci. Comput. 30(6), 3197–3219 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 33.
    Wild, S.M., Shoemaker, C.A.: Global convergence of radial basis function trust-region algorithms for derivative-free optimization. SIGEST article. SIAM Rev. 55(2), 349–371 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 34.
    Ye, K.Q., Li, W., Sudjianto, A.: Algorithmic construction of optimal symmetric latin hypercube designs. J. Stat. Plan. Infer. 90, 145–159 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesChongqing Normal UniversityChongqingChina

Personalised recommendations