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On the search of the shape parameter in radial basis functions using univariate global optimization methods

Author

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  • R. Cavoretto

    (University of Torino)

  • A. Rossi

    (University of Torino)

  • M. S. Mukhametzhanov

    (University of Calabria
    Lobachevsky Nizhni Novgorod State University)

  • Ya. D. Sergeyev

    (University of Calabria
    Lobachevsky Nizhni Novgorod State University)

Abstract

In this paper we consider the problem of finding an optimal value of the shape parameter in radial basis function interpolation. In particular, we propose the use of a leave-one-out cross validation (LOOCV) technique combined with univariate global optimization methods, which involve strategies of global optimization with pessimistic improvement (GOPI) and global optimization with optimistic improvement (GOOI). This choice is carried out to overcome serious issues of commonly used optimization routines that sometimes result in shape parameter values are not globally optimal. New locally-biased versions of geometric and information Lipschitz global optimization algorithms are presented. Numerical experiments and applications to real-world problems show a promising performance and efficacy of the new algorithms, called LOOCV-GOPI and LOOCV-GOOI, in comparison with their direct competitors.

Suggested Citation

  • R. Cavoretto & A. Rossi & M. S. Mukhametzhanov & Ya. D. Sergeyev, 2021. "On the search of the shape parameter in radial basis functions using univariate global optimization methods," Journal of Global Optimization, Springer, vol. 79(2), pages 305-327, February.
    • Handle: RePEc:spr:jglopt:v:79:y:2021:i:2:d:10.1007_s10898-019-00853-3
    DOI: 10.1007/s10898-019-00853-3
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    References listed on IDEAS

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    1. Yao, Guangming & Duo, Jia & Chen, C.S. & Shen, L.H., 2015. "Implicit local radial basis function interpolations based on function values," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 91-102.
    2. Biazar, Jafar & Hosami, Mohammad, 2017. "An interval for the shape parameter in radial basis function approximation," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 131-149.
    3. J. Calvin & A. Žilinskas, 1999. "On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions," Journal of Optimization Theory and Applications, Springer, vol. 102(3), pages 479-495, September.
    4. Antanas Žilinskas, 2010. "On similarities between two models of global optimization: statistical models and radial basis functions," Journal of Global Optimization, Springer, vol. 48(1), pages 173-182, September.
    5. Grishagin, Vladimir & Israfilov, Ruslan & Sergeyev, Yaroslav, 2018. "Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 270-280.
    6. Kvasov, Dmitri E. & Mukhametzhanov, Marat S., 2018. "Metaheuristic vs. deterministic global optimization algorithms: The univariate case," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 245-259.
    7. Yaroslav D. Sergeyev & Marat S. Mukhametzhanov & Dmitri E. Kvasov & Daniela Lera, 2016. "Derivative-Free Local Tuning and Local Improvement Techniques Embedded in the Univariate Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 186-208, October.
    8. Sergeyev, Yaroslav D. & Kvasov, Dmitri E. & Mukhametzhanov, Marat S., 2017. "Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 96-109.
    9. Anatoly Zhigljavsky & Antanas Žilinskas, 2008. "Stochastic Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-74740-8, September.
    10. Yaroslav D. Sergeyev & Dmitri E. Kvasov & Marat S. Mukhametzhanov, 2016. "On the Least-Squares Fitting of Data by Sinusoids," Springer Optimization and Its Applications, in: Panos M. Pardalos & Anatoly Zhigljavsky & Julius Žilinskas (ed.), Advances in Stochastic and Deterministic Global Optimization, pages 209-226, Springer.
    11. Jaroslav Fowkes & Nicholas Gould & Chris Farmer, 2013. "A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions," Journal of Global Optimization, Springer, vol. 56(4), pages 1791-1815, August.
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