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

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Abstract

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.

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References

  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. Loh, W.-L., Lam, T.-K.: Estimating structured correlation matrices in smooth Gaussian random field models. Ann. Stat. 28, 880–904 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mockus, J.: Bayesian Approach to Global Optimization. Kluwer Academic Publishers, Dordrecht (1988)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sacks, J., Schiller, S.B., Welch, W.J.: Designs for computer experiments. Technometrics 31(1), 41–47 (1989)

    Article  MathSciNet  Google Scholar 

  6. Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–423 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wanting, X., Stein, M.L.: Maximum likelihood estimation for smooth Gaussian random field model. SIAM/ASA Uncertain. Quantif. 5, 138–175 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)

    MATH  Google Scholar 

  9. Žilinskas, A.: Optimization of one-dimensional multimodal functions, Algorithm AS133. J. R. Stat. Soc. Ser. C 23, 367–385 (1978)

    MATH  Google Scholar 

  10. Žilinskas, A.: A statistical model-based algorithm for black-box multi-objective optimisation. Int. J. Syst. Sci. 45(1), 82–92 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Žilinskas, A., Zhigljavsky, A.: Stochastic global optimization: a review on the occasion of 25 years of Informatica. Informatica 27(2), 229–256 (2016)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Institute of Data Science and Digital Technologies, Vilnius University, Akademijos 4, 08663, Vilnius, Lithuania

    Antanas Žilinskas

  2. School of Mathematics, Cardiff University, Cardiff, CF24 1AG, UK

    Anatoly Zhigljavsky

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  1. Anatoly Zhigljavsky
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  2. Antanas Žilinskas
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Correspondence to Antanas Žilinskas.

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Zhigljavsky, A., Žilinskas, A. Selection of a covariance function for a Gaussian random field aimed for modeling global optimization problems. Optim Lett 13, 249–259 (2019). https://doi.org/10.1007/s11590-018-1372-5

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