Journal of Global Optimization

, Volume 41, Issue 3, pp 447–464

An adaptive radial basis algorithm (ARBF) for expensive black-box global optimization

Article

Abstract

Powerful response surface methods based on kriging and radial basis function (RBF) interpolation have been developed for expensive, i.e. computationally costly, global nonconvex optimization. We have implemented some of these methods in the solvers rbfSolve and EGO in the TOMLAB Optimization Environment (http://www.tomopt.com/tomlab/). In this paper we study algorithms based on RBF interpolation. The practical performance of the RBF algorithm is sensitive to the initial experimental design, and to the static choice of target values. A new adaptive radial basis interpolation (ARBF) algorithm, suitable for parallel implementation, is presented. The algorithm is described in detail and its efficiency is analyzed on the standard test problem set of Dixon–Szegö. Results show that it outperforms the published results of rbfSolve and several other solvers.

Keywords

Global optimization Expensive function CPU-intensive Costly function Mixed-integer Nonconvex Software Black-box Derivative-free Response surface Radial basis functions Surrogate model Response surface Splines 

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References

  1. 1.
    Björkman M. and Holmström K. (2000). Global optimization of costly nonconvex functions using radial basis functions. Optim. Eng. 1(4): 373–397 CrossRefGoogle Scholar
  2. 2.
    Dixon, L.C.W., Szegö, G.P.: The global optimisation problem: an introduction. In: Dixon, L., Szego G. (eds.) Toward Global Optimization, vol. 2, pp. 1–15. New York (1978)Google Scholar
  3. 3.
    Gutmann, H.-M.: A Radial Basis for Function Method for Global Optimization. In: Talk at IFIP TC7 Conference, Cambridge (1999a)Google Scholar
  4. 4.
    Gutmann, H.-M.: A radial basis function method for global optimization. Technical Report DAMTP 1999/NA22. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England (1999b)Google Scholar
  5. 5.
    Gutmann H.-M. (2001a). A radial basis function method for global optimization. J. Global Optim. 19: 201–227 CrossRefGoogle Scholar
  6. 6.
    Gutmann, H.-M.: Radial basis function methods for global optimization. Doctoral Thesis, Department of Numerical Analysis, Cambridge University, Cambridge, UK (2001b)Google Scholar
  7. 7.
    Holmström, K., Edvall, M.M.: Chapter 19: The TOMLAB optimization environment. In: Josef~Kallrath, L.G. (ed.) Modeling Languages in Mathematical Optimization. Boston/Dordrecht/London (2004)Google Scholar
  8. 8.
    Holmström, K., Quttineh, N.-H., Edvall, M.M.: An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization. Optim. Eng. (Under review) (2007)Google Scholar
  9. 9.
    Horst R. and Pardalos P.M. (1995). Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht, Boston, London Google Scholar
  10. 10.
    Huyer W. and Neumaier A. (1999). Global optimization by multilevel coordinate Search. J. Global Optim. 14: 331–355 CrossRefGoogle Scholar
  11. 11.
    Jones D.R. (2002). A taxonomy of global optimization methods based on response surfaces. J. Global Optim. 21: 345–383 CrossRefGoogle Scholar
  12. 12.
    Jones D.R., Perttunen C.D. and Stuckman B.E. (1993). Lipschitzian optimization without the Lipschitz constant. J. optim. Theory Appl. 79(1): 157–181 CrossRefGoogle Scholar
  13. 13.
    Jones D.R., Schonlau M. and Welch W.J. (1998). Efficient global optimization of expensive black-box functions. J. Global Optim. 13: 455–492 CrossRefGoogle Scholar
  14. 14.
    Powell, M.J.D.: The theory of radial basis function approximation in 1990. In: Light, W. (ed.) Advances in Numerical Analysis, vol. 2: Wavelets, Subdivision Algorithms and Radial Basis Functions, pp. 105–210. Oxford University Press (1992)Google Scholar
  15. 15.
    Powell M.J.D. (1999). Recent research at Cambridge on radial basis functions. In: Buhmann, M.D., Felten, M., Mache, D. and Müller, M.W. (eds) New Developments in Approximation Theory., pp 215–232. Birkhäuser, Basel Google Scholar
  16. 16.
    Regis R.G. and Shoemaker C.A. (2005). Constrained global optimization of expensive black box functions using radial basis functions. J. Global Optim. 31(1): 153–171 CrossRefGoogle Scholar
  17. 17.
    Törn, A., Zilinskas, A.: Lecture Notes in Computer Science, vol. 350. Berlin Heidelberg (1987)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsMälardalen UniversityVästeråsSweden

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