Optimization Letters

, Volume 13, Issue 2, pp 249–259 | Cite as

Selection of a covariance function for a Gaussian random field aimed for modeling global optimization problems

  • Anatoly Zhigljavsky
  • Antanas ŽilinskasEmail author
Original Paper


Bayesian approach is actively used to develop global optimization algorithms aimed at expensive black box functions. One of the challenges in this approach is the selection of an appropriate model for the objective function. Normally, a Gaussian random field is chosen as a theoretical model. However, the problem of estimation of parameters, using objective function values, is not thoroughly researched. In this paper, we consider the behavior of the maximum likelihood estimators of parameters of the homogeneous isotropic Gaussian random field with squared exponential covariance function. We also compare properties of exponential covariance function models.


Global optimization Bayesian approach Statistical models Gaussian random fields Parameters’ estimation 



The work of A.Zilinskas was supported by the Research Council of Lithuania under Grant No. P-MIP-17-61. The work of A.Zhigljavsky was supported by a grant of Crimtant Holding Limited.


  1. 1.
    Knowles, J., Corne, D., Reynolds, A.: Noisy multiobjective optimization on a budget of 250 evaluations. In: Ehrgott, M., et al. (eds.) Lecture Notes in Computer Science, vol. 5467, pp. 36–50. Springer, New York (2009)Google Scholar
  2. 2.
    Loh, W.-L., Lam, T.-K.: Estimating structured correlation matrices in smooth Gaussian random field models. Ann. Stat. 28, 880–904 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Mockus, J.: Bayesian Approach to Global Optimization. Kluwer Academic Publishers, Dordrecht (1988)zbMATHGoogle Scholar
  4. 4.
    Pepelyshev, A.: Fixed-domain asymtotics of the maximum likelihood estiomator and the gaussian process approach for deterministic models. Stat. Methodol. 8(4), 356–362 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sacks, J., Schiller, S.B., Welch, W.J.: Designs for computer experiments. Technometrics 31(1), 41–47 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–423 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wanting, X., Stein, M.L.: Maximum likelihood estimation for smooth Gaussian random field model. SIAM/ASA Uncertain. Quantif. 5, 138–175 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)zbMATHGoogle Scholar
  9. 9.
    Žilinskas, A.: Optimization of one-dimensional multimodal functions, Algorithm AS133. J. R. Stat. Soc. Ser. C 23, 367–385 (1978)zbMATHGoogle Scholar
  10. 10.
    Žilinskas, A.: A statistical model-based algorithm for black-box multi-objective optimisation. Int. J. Syst. Sci. 45(1), 82–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Žilinskas, A., Zhigljavsky, A.: Stochastic global optimization: a review on the occasion of 25 years of Informatica. Informatica 27(2), 229–256 (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Žilinskas, A., Žilinskas, J.: Parallel hybrid algorithm for global optimization of problems occurring in MDS-based visualization. Comput. Math. Appl. 52, 211–224 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Data Science and Digital TechnologiesVilnius UniversityVilniusLithuania
  2. 2.School of MathematicsCardiff UniversityCardiffUK

Personalised recommendations