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International Conference on Optimization and Applications

OPTIMA 2019: Optimization and Applications pp 326–339Cite as

An Experimental Study of Univariate Global Optimization Algorithms for Finding the Shape Parameter in Radial Basis Functions

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1145))

Abstract

In this contribution, an interpolation problem using radial basis functions is considered. A recently proposed approach for the search of the optimal value of the shape parameter is studied. The approach consists of using global optimization algorithms to minimize the error function obtained using a leave-one-out cross validation (LOOCV) technique, which is commonly used for solving machine learning problems. In this paper, the proposed approach is studied experimentally on classes of randomly generated test problems using the GKLS-generator, which is widely used for testing global optimization algorithms. The experimental study on classes of randomly generated test problems is very important from the practical point of view, since results show the behavior of the algorithms for solving not a single test problem, but the whole class with controllable difficulty, which is the main property of the GKLS-generator. The obtained results are relevant, since the experiments have been carried out on 200 randomized test problems, and show that the algorithms are efficient for solving difficult real-life problems demonstrating a promising behavior.

The work of M. S. Mukhametzhanov was supported by the project “Smart Electronic Invoices Accounting” - SELINA CUP: J28C17000160006 (POR CALABRIA FESR-FSE 2014–2020) and by the INdAM-GNCS funding “Giovani Ricercatori 2018–2019”. The work of R. Cavoretto and A. De Rossi was partially supported by the Department of Mathematics “Giuseppe Peano” of the University of Torino via Project 2019 “Mathematics for applications” and by the INdAM–GNCS Project 2019 “Kernel-based approximation, multiresolution and subdivision methods and related applications”.

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Notes

  1. 1.

    Traditionally, terms “simple” and “difficult” are related to the difficulty of locating the global minimizer and are used for testing global optimization algorithms and not interpolation methods. In this paper, these terms are used only to distinguish these two classes and not to indicate the difficulty of the interpolation problem.

References

  1. Barkalov, K., Gergel, V., Lebedev, I.: Solving global optimization problems on GPU cluster. In: Simos, T.E. (ed.) AIP Conference Proceedings, vol. 1738, p. 400006 (2016)

    Google Scholar 

  2. Barkalov, K., Strongin, R.: Solving a set of global optimization problems by the parallel technique with uniform convergence. J. Global Optim. 71(1), 21–36 (2018)

    Article  MathSciNet  Google Scholar 

  3. Cavoretto, R., De Rossi, A.: A trivariate interpolation algorithm using a cube-partition searching procedure. SIAM J. Sci. Comput. 37, A1891–A1908 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cavoretto, R., De Rossi, A., Mukhametzhanov, M.S., Sergeyev, Y.D.: On the search of the shape parameter in radial basis functions using univariate global optimization methods. J. Glob. Optim. (2019, in press). https://doi.org/10.1007/s10898-019-00853-3

  5. Fasshauer, G.E.: Meshfree Approximation Methods with Matlab, Interdisciplinary Mathematical Sciences, vol. 6. World Scientific, Singapore (2007)

    Book  Google Scholar 

  6. Fasshauer, G.E.: Positive definite kernels: past, present and future. Dolomites Res. Notes Approx. 4, 21–63 (2011)

    Google Scholar 

  7. Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33, 869–892 (2011)

    Article  MathSciNet  Google Scholar 

  8. Gaviano, M., Kvasov, D.E., Lera, D., Sergeyev, Y.D.: Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29(4), 469–480 (2003)

    Article  MathSciNet  Google Scholar 

  9. Gergel, V.P., Grishagin, V.A., Israfilov, R.A.: Local tuning in nested scheme of global optimization. Procedia Comput. Sci. 51, 865–874 (2015)

    Article  Google Scholar 

  10. Gergel, V.P., Kuzmin, M.I., Solovyov, N.A., Grishagin, V.A.: Recognition of surface defects of cold-rolling sheets based on method of localities. Int. Rev. Autom. Control 8(1), 51–55 (2015)

    Google Scholar 

  11. Gillard, J.W., Zhigljavsky, A.A.: Stochastic algorithms for solving structured low-rank matrix approximation problems. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 70–88 (2015)

    Article  MathSciNet  Google Scholar 

  12. Golbabai, A., Mohebianfar, E., Rabiei, H.: On the new variable shape parameter strategies for radial basis functions. Comput. Appl. Math. 34, 691–704 (2015)

    Article  MathSciNet  Google Scholar 

  13. Grishagin, V.A., Israfilov, R.A., Sergeyev, Y.D.: Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes. Appl. Math. Comput. 318, 270–280 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Grishagin, V.A., Sergeyev, Y.D., Strongin, R.G.: Parallel characteristic algorithms for solving problems of global optimization. J. Global Optim. 10(2), 185–206 (1997)

    Article  MathSciNet  Google Scholar 

  15. Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization, vol. 1. Kluwer Academic Publishers, Dordrecht (1995)

    MATH  Google Scholar 

  16. Khamisov, O.V., Posypkin, M.: Univariate global optimization with point-dependent Lipschitz constants. In: AIP Conference Proceedings, vol. 2070, p. 020051. AIP Publishing (2019)

    Google Scholar 

  17. Khamisov, O., Posypkin, M., Usov, A.: Piecewise linear bounding functions for univariate global optimization. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds.) OPTIMA 2018. CCIS, vol. 974, pp. 170–185. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10934-9_13

    Chapter  Google Scholar 

  18. Kvasov, D.E., Mukhametzhanov, M.S.: One-dimensional global search: nature-inspired vs. Lipschitz methods. In: Simos, T.E. (ed.) AIP Conference Proceedings, vol. 1738, p. 400012 (2016)

    Google Scholar 

  19. Kvasov, D.E., Mukhametzhanov, M.S.: Metaheuristic vs. deterministic global optimization algorithms: the univariate case. Appl. Math. Computat. 318, 245–259 (2018)

    Article  MathSciNet  Google Scholar 

  20. Kvasov, D.E., Sergeyev, Y.D.: Univariate geometric Lipschitz global optimization algorithms. Numer. Algebra Control Optim. 2(1), 69–90 (2012)

    Article  MathSciNet  Google Scholar 

  21. Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. SpringerBriefs in Optimization. Springer, New York (2014). https://doi.org/10.1007/978-1-4614-9093-7

    Book  MATH  Google Scholar 

  22. Piyavskij, S.A.: An algorithm for finding the absolute extremum of a function. USSR Comput. Math. Math. Phys. 12(4), 57–67 (1972). (in Russian) Zh. Vychisl. Mat. Mat. Fiz. 12(4), pp. 888–896 (1972)

    Google Scholar 

  23. Ratz, D.: A nonsmooth global optimization technique using slopes: the one-dimensional case. J. Global Optim. 14(4), 365–393 (1999)

    Article  MathSciNet  Google Scholar 

  24. Rippa, S.: An algorithm for selecting a good value for the parameter \(c\) in radial basis function interpolation. Adv. Comput. Math. 11, 193–210 (1999)

    Article  MathSciNet  Google Scholar 

  25. Sergeyev, Y.D., Daponte, P., Grimaldi, D., Molinaro, A.: Two methods for solving optimization problems arising in electronic measurements and electrical engineering. SIAM J. Optim. 10(1), 1–21 (1999)

    Article  MathSciNet  Google Scholar 

  26. Sergeyev, Y.D., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16(3), 910–937 (2006)

    Article  MathSciNet  Google Scholar 

  27. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: On the least-squares fitting of data by sinusoids. In: Pardalos, P.M., Zhigljavsky, A., Žilinskas, J. (eds.) Advances in Stochastic and Deterministic Global Optimization. SOIA, vol. 107, pp. 209–226. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29975-4_11

    Chapter  Google Scholar 

  28. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: Emmental-type GKLS-based multiextremal smooth test problems with non-linear constraints. In: Battiti, R., Kvasov, D.E., Sergeyev, Y.D. (eds.) LION 2017. LNCS, vol. 10556, pp. 383–388. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-69404-7_35

    Chapter  Google Scholar 

  29. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms. Math. Comput. Simul. 141, 96–109 (2017)

    Article  MathSciNet  Google Scholar 

  30. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales. Commun. Nonlinear Sci. Numer. Simul. 59, 319–330 (2018)

    Article  MathSciNet  Google Scholar 

  31. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget. Sci. Rep. 8, 453 (2018)

    Article  Google Scholar 

  32. Sergeyev, Y.D., Mukhametzhanov, M.S., Kvasov, D.E., Lera, D.: Derivative-free local tuning and local improvement techniques embedded in the univariate global optimization. J. Optim. Theory Appl. 171(1), 186–208 (2016)

    Article  MathSciNet  Google Scholar 

  33. Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000). 3rd ed. by Springer (2014)

    Chapter  Google Scholar 

  34. Uddin, M.: On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method. Appl. Math. Model. 38, 135–144 (2014)

    Article  MathSciNet  Google Scholar 

  35. Zhigljavsky, A.A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008). https://doi.org/10.1007/978-0-387-74740-8

    Book  MATH  Google Scholar 

  36. Žilinskas, A.: On similarities between two models of global optimization: statistical models and radial basis functions. J. Global Optim. 48(1), 173–182 (2010)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Calabria, Via Pietro Bucci 42C, 87036, Rende, CS, Italy

    Marat S. Mukhametzhanov

  2. Lobachevsky State University, Gagarin Avenue 23, 603950, Nizhny Novgorod, Russia

    Marat S. Mukhametzhanov

  3. Department of Mathematics “Giuseppe Peano”, University of Torino, via Carlo Alberto 10, 10123, Turin, Italy

    Roberto Cavoretto & Alessandra De Rossi

Authors
  1. Marat S. Mukhametzhanov
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  2. Roberto Cavoretto
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  3. Alessandra De Rossi
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Correspondence to Marat S. Mukhametzhanov .

Editor information

Editors and Affiliations

  1. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  2. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

  4. FRC CSC RAS, Moscow, Russia

    Mikhail Posypkin

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Mukhametzhanov, M.S., Cavoretto, R., De Rossi, A. (2020). An Experimental Study of Univariate Global Optimization Algorithms for Finding the Shape Parameter in Radial Basis Functions. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2019. Communications in Computer and Information Science, vol 1145. Springer, Cham. https://doi.org/10.1007/978-3-030-38603-0_24

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  • DOI: https://doi.org/10.1007/978-3-030-38603-0_24

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