Skip to main content
Springer Nature Link
Log in

On Acceleration of Derivative-Free Univariate Lipschitz Global Optimization Methods

  • Conference paper
  • First Online:
Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

  • 767 Accesses

Abstract

Univariate box-constrained global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous and multiextremal. It is assumed that its analytical representation is unknown (the function is given as a “black-box”) and even one its evaluation is a computationally expensive procedure. Geometric and information statistical frameworks for construction of global optimization algorithms are discussed. Several powerful acceleration techniques are described and a number of methods of both classes is constructed by mixing the introduced acceleration ideas. Numerical experiments executed on broad test classes taken from the literature show advantages of the presented techniques with respect to their direct competitors.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Barkalov, K.A., Gergel, V.P.: Parallel global optimization on GPU. J. Glob. Optim. 66(1), 3–20 (2016)

    Article  MathSciNet  Google Scholar 

  2. Calvin, J.M., Žilinskas, A.: One-dimensional global optimization for observations with noise. Comput. Math. Appl. 50(1–2), 157–169 (2005)

    Article  MathSciNet  Google Scholar 

  3. Daponte, P., Grimaldi, D., Molinaro, A., Sergeyev, Y.D.: Fast detection of the first zero-crossing in a measurement signal set. Measurement 19(1), 29–39 (1996)

    Article  Google Scholar 

  4. Floudas, C.A., Pardalos, P.M.: State of the Art in Global Optimization. Kluwer Academic Publishers, Dordrecht (1996)

    Book  Google Scholar 

  5. 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 

  6. Grishagin, V.A., Israfilov, R.A., Sergeyev, Y.D.: Comparative efficiency of dimensionality reduction schemes in global optimization. In: Proceedings of the 2nd International Conference on “Numerical Computations: Theory and Algorithms”, vol. 1776, p. 060011. AIP Publishing, New York (2016). https://doi.org/10.1063/1.4965345

  7. 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 

  8. Hansen, P., Jaumard, B., Lu, S.H.: Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison. Math. Program. 55(1–3), 273–292 (1992)

    Article  MathSciNet  Google Scholar 

  9. Kvasov, D.E., Pizzuti, C., Sergeyev, Y.D.: Local tuning and partition strategies for diagonal GO methods. Numer. Math. 94(1), 93–106 (2003)

    Article  MathSciNet  Google Scholar 

  10. Kvasov, D.E., Sergeyev, Y.D.: Deterministic approaches for solving practical black-box global optimization problems. Adv. Eng. Softw. 80, 58–66 (2015)

    Article  Google Scholar 

  11. Lera, D., Sergeyev, Y.D.: An information global minimization algorithm using the local improvement technique. J. Glob. Optim. 48(1), 99–112 (2010)

    Article  MathSciNet  Google Scholar 

  12. Lera, D., Sergeyev, Y.D.: Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives. SIAM J. Optim. 1(23), 508–529 (2013)

    Article  MathSciNet  Google Scholar 

  13. Modorskii, V.Y., Gaynutdinova, D.F., Gergel, V.P., Barkalov, K.A.: Optimization in design of scientific products for purposes of cavitation problems. In: Proceedings of the International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2015), vol. 1738, p. 400013. AIP Publishing, New York (2016). https://doi.org/10.1063/1.4952201

  14. Paulavičius, R., Sergeyev, Y.D., Kvasov, D.E., Žilinskas, J.: Globally-biased DISIMPL algorithm for expensive global optimization. J. Glob. Optim. 59(2–3), 545–567 (2014)

    Article  MathSciNet  Google Scholar 

  15. Pintér, J.D.: Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications). Kluwer, Dordrecht (1996)

    Book  Google Scholar 

  16. Piyavskij, S.A.: An algorithm for finding the absolute extremum of a function. USSR Comput. Math. Math. Phys. 12(4), 57–67 (1972)

    Article  Google Scholar 

  17. Sergeyev, Y.D.: An information global optimization algorithm with local tuning. SIAM J. Optim. 5(4), 858–870 (1995)

    Article  MathSciNet  Google Scholar 

  18. Sergeyev, Y.D.: A one-dimensional deterministic global minimization algorithm. Comput. Math. Math. Phys. 35(5), 705–717 (1995)

    MathSciNet  Google Scholar 

  19. Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81(1), 127–146 (1998)

    Article  MathSciNet  Google Scholar 

  20. 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 

  21. Sergeyev, Y.D., Grishagin, V.A.: A parallel method for finding the global minimum of univariate functions. J. Optimiz. Theor. Appl. 80(3), 513–536 (1994)

    Article  MathSciNet  Google Scholar 

  22. Sergeyev, Y.D., Kvasov, D.E.: Deterministic Global Optimization: An Introduction to the Diagonal Approach. Springer, New York (2017)

    Book  Google Scholar 

  23. 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. 59, 319–330 (2018)

    Article  MathSciNet  Google Scholar 

  24. Sergeyev, Y.D., Kvasov, D.E., Mukhametzhanov, M.S.: On the efficiency of nature-inspired metaheuristics in expensive global optimization with limited budget. Nat. Sci. Rep. 8, Article 453 (2018). https://doi.org/10.1038/s41598-017-18940-4

  25. 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. Optimiz. Theor. Appl. 171(1), 319–330 (2016)

    Article  MathSciNet  Google Scholar 

  26. Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York (2013)

    Book  Google Scholar 

  27. Strongin, R.G.: On the convergence of an algorithm for finding a global extremum. Eng. Cybern. 11, 549–555 (1973)

    MathSciNet  Google Scholar 

  28. Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The work of M.S. Mukhametzhanov was supported by the INdAM-GNCS funding “Giovani Ricercatori 2018–2019”.

Author information

Authors and Affiliations

  1. University of Calabria, Rende, CS, Italy

    Dmitri E. Kvasov, Marat S. Mukhametzhanov, Maria Chiara Nasso & Yaroslav D. Sergeyev

  2. Lobachevsky State University, Nizhni Novgorod, Russia

    Dmitri E. Kvasov, Marat S. Mukhametzhanov & Yaroslav D. Sergeyev

Authors
  1. Dmitri E. Kvasov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marat S. Mukhametzhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maria Chiara Nasso
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yaroslav D. Sergeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitri E. Kvasov .

Editor information

Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kvasov, D.E., Mukhametzhanov, M.S., Nasso, M.C., Sergeyev, Y.D. (2020). On Acceleration of Derivative-Free Univariate Lipschitz Global Optimization Methods. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_38

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-40616-5_38

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-40615-8

  • Online ISBN: 978-3-030-40616-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics