IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v79y2021i2d10.1007_s10898-019-00853-3.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00853-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-019-00853-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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, June.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Martín Alejandro Valencia-Ponce & Esteban Tlelo-Cuautle & Luis Gerardo de la Fraga, 2021. "Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
    2. Petr Fedoseev & Artur Karimov & Vincent Legat & Denis Butusov, 2022. "Preference and Stability Regions for Semi-Implicit Composition Schemes," Mathematics, MDPI, vol. 10(22), pages 1-13, November.
    3. Zhang, Yuhui & Rabczuk, Timon & Lin, Ji & Lu, Jun & Chen, C.S., 2024. "Numerical simulations of two-dimensional incompressible Navier-Stokes equations by the backward substitution projection method," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    4. Zheng, Sanpeng & Feng, Renzhong, 2023. "A variable projection method for the general radial basis function neural network," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    5. Chen, Chuin-Shan & Noorizadegan, Amir & Young, D.L. & Chen, C.S., 2023. "On the selection of a better radial basis function and its shape parameter in interpolation problems," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    6. Li, Yang & Liu, Dejun & Yin, Zhexu & Chen, Yun & Meng, Jin, 2023. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    7. Noorizadegan, A. & Young, D.L. & Hon, Y.C. & Chen, C.S., 2024. "Power-enhanced residual network for function approximation and physics-informed inverse problems," Applied Mathematics and Computation, Elsevier, vol. 480(C).
    8. Samaneh Mokhtari & Ali Mesforush & Reza Mokhtari & Rahman Akbari & Clemens Heitzinger, 2023. "Solving Stochastic Nonlinear Poisson-Boltzmann Equations Using a Collocation Method Based on RBFs," Mathematics, MDPI, vol. 11(9), pages 1-13, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Daniela Lera & Yaroslav D. Sergeyev, 2018. "GOSH: derivative-free global optimization using multi-dimensional space-filling curves," Journal of Global Optimization, Springer, vol. 71(1), pages 193-211, May.
    3. 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.
    4. Konstantin Barkalov & Irek Gubaydullin & Evgeny Kozinov & Ilya Lebedev & Roza Faskhutdinova & Azamat Faskhutdinov & Leniza Enikeeva, 2022. "On Solving the Problem of Finding Kinetic Parameters of Catalytic Isomerization of the Pentane-Hexane Fraction Using a Parallel Global Search Algorithm," Mathematics, MDPI, vol. 10(19), pages 1-13, October.
    5. Yong Wang & Kunzhao Wang & Gaige Wang, 2022. "Neural Network Algorithm with Dropout Using Elite Selection," Mathematics, MDPI, vol. 10(11), pages 1-17, May.
    6. 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.
    7. Albertas Gimbutas & Antanas Žilinskas, 2018. "An algorithm of simplicial Lipschitz optimization with the bi-criteria selection of simplices for the bi-section," Journal of Global Optimization, Springer, vol. 71(1), pages 115-127, May.
    8. Juan Li & Dan-dan Xiao & Hong Lei & Ting Zhang & Tian Tian, 2020. "Using Cuckoo Search Algorithm with Q -Learning and Genetic Operation to Solve the Problem of Logistics Distribution Center Location," Mathematics, MDPI, vol. 8(2), pages 1-32, January.
    9. 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.
    10. Lera, Daniela & Posypkin, Mikhail & Sergeyev, Yaroslav D., 2021. "Space-filling curves for numerical approximation and visualization of solutions to systems of nonlinear inequalities with applications in robotics," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    11. Ziadi, Raouf & Bencherif-Madani, Abdelatif & Ellaia, Rachid, 2020. "A deterministic method for continuous global optimization using a dense curve," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 62-91.
    12. Andrey Pepelyshev & Anatoly Zhigljavsky & Antanas Žilinskas, 2018. "Performance of global random search algorithms for large dimensions," Journal of Global Optimization, Springer, vol. 71(1), pages 57-71, May.
    13. Anis Fradi & Chafik Samir & Ines Adouani, 2024. "A New Bayesian Approach to Global Optimization on Parametrized Surfaces in $$\mathbb {R}^{3}$$ R 3," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1077-1100, September.
    14. Vasiliy V. Grigoriev & Petr N. Vabishchevich, 2021. "Bayesian Estimation of Adsorption and Desorption Parameters for Pore Scale Transport," Mathematics, MDPI, vol. 9(16), pages 1-16, August.
    15. Jonathan Gillard & Anatoly Zhigljavsky, 2013. "Optimization challenges in the structured low rank approximation problem," Journal of Global Optimization, Springer, vol. 57(3), pages 733-751, November.
    16. Assari, Pouria & Asadi-Mehregan, Fatemeh & Cuomo, Salvatore, 2019. "A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 157-172.
    17. Sergey S. Ketkov & Oleg A. Prokopyev & Lisa M. Maillart, 2023. "Planning of life-depleting preventive maintenance activities with replacements," Annals of Operations Research, Springer, vol. 324(1), pages 1461-1483, May.
    18. Renato Bruni & Fabio Celani, 2019. "Combining Global and Local Strategies to Optimize Parameters in Magnetic Spacecraft Control via Attitude Feedback," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 997-1014, June.
    19. Linas Stripinis & Remigijus Paulavičius, 2023. "Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions," Mathematics, MDPI, vol. 11(13), pages 1-19, June.
    20. Cheng-Yu Ku & Jing-En Xiao & Chih-Yu Liu, 2020. "A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations," Mathematics, MDPI, vol. 8(2), pages 1-22, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:79:y:2021:i:2:d:10.1007_s10898-019-00853-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.