Multidimensional Global Search Using Numerical Estimations of Minimized Function Derivatives and Adaptive Nested Optimization Scheme

  • Victor GergelEmail author
  • Alexey Goryachikh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11974)


This paper proposes a novel approach to the solution of time-consuming multivariate multiextremal optimization problems. This approach is based on integrating the global search method using derivatives of minimized functions and the nested scheme for dimensionality reduction. In contrast with related works novelty is that derivative values are calculated numerically and the dimensionality reduction scheme is generalized for adaptive use of the search information. The obtained global optimization method demonstrates a good performance, which has been confirmed by numerical experiments.


Multiextremal optimization Global search algorithms Lipschitz condition Numerical estimations of derivative values Dimensionality reduction Numerical experiments 



The reported study was funded by the RFBR under research project No. 19-07-00242.


  1. 1.
    Strongin, R., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms, 3rd edn. Kluwer Academic Publishers, Dordrecht (2014)zbMATHGoogle Scholar
  2. 2.
    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)CrossRefGoogle Scholar
  3. 3.
    Floudas, C., Pardalos, M.: Recent Advances in Global Optimization. Princeton University Press, Princeton (2016). Scholar
  4. 4.
    Pardalos, M., Zhigljavsky, A., Žilinskas, J.: Advances in Stochastic and Deterministic Global Optimization. Springer, Cham (2016). Scholar
  5. 5.
    Famularo, D., Pugliese, P., Sergeyev, Y.D.: A global optimization technique for checking parametric robustness. Automatica 35, 1605–1611 (1999). Scholar
  6. 6.
    Modorskii, V., Gaynutdinova, D., Gergel, V., Barkalov, K.: Optimization in design of scientific products for purposes of cavitation problems. AIP Conf. Proc. 1738, 400013 (2016). Scholar
  7. 7.
    Piyavskij, S.: An algorithm for finding the absolute extremum of a function. Comput. Math. Math. Phys. 12, 57–67 (1972). (in Russian)CrossRefGoogle Scholar
  8. 8.
    Shubert, B.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9, 379–388 (1972). Scholar
  9. 9.
    Strongin, R.: On the convergence of an algorithm for finding a global extremum. Eng. Cybern. 11, 549–555 (1973)MathSciNetGoogle Scholar
  10. 10.
    Gergel, V.: A method of using derivatives in the minimization of multiextremum functions. Comput. Math. Math. Phys. 36, 729–742 (1996). (In Russian)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81, 127–146 (1998). Scholar
  12. 12.
    Gergel, V., Goryachih, A.: Global optimization using numerical approximations of derivatives. In: Battiti, R., Kvasov, D.E., Sergeyev, Y.D. (eds.) LION 2017. LNCS, vol. 10556, pp. 320–325. Springer, Cham (2017). Scholar
  13. 13.
    Sergeyev, Y.D., Strongin, R., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York (2013). Scholar
  14. 14.
    Gergel, V.P., Strongin, R.G.: Parallel computing for globally optimal decision making. In: Malyshkin, V.E. (ed.) PaCT 2003. LNCS, vol. 2763, pp. 76–88. Springer, Heidelberg (2003). Scholar
  15. 15.
    Barkalov, K., Gergel, V., Lebedev, I.: Solving global optimization problems on GPU cluster. AIP Conf. Proc. 1738, 400006 (2016). Scholar
  16. 16.
    Sergeyev, Y.D., Kvasov, D.E.: A deterministic global optimization using smooth diagonal auxiliary functions. Commun. Nonlinear Sci. Numer. Simul. 21, 99–111 (2015). Scholar
  17. 17.
    Lera, D., Sergeyev, Y.D.: Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants. Commun. Nonlinear Sci. Numer. Simul. 23, 328–342 (2015). Scholar
  18. 18.
    Gergel, V., Grishagin, V., Gergel, A.: Adaptive nested optimization scheme for multidimensional global search. J. Glob. Optim. 6, 35–51 (2015). Scholar
  19. 19.
    Gergel, V., Grishagin, V., Israfilov, R.: Local tuning in nested scheme of global optimization. Procedia Comput. Sci. 51, 865–874 (2015). Scholar
  20. 20.
    Grishagin, V., Israfilov, R., Sergeyev, Y.D.: Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction scheme. Appl. Math. Comput. 318, 270–280 (2018). Scholar
  21. 21.
    Sergeyev, Y.D., Mukhametzhanov, M.S., Kvasov, D.E.: Operational zones for comparing metaheuristic and deterministic one-dimensional global optimization algorithms. Math. Comput. Simul. 141, 96–109 (2017). Scholar
  22. 22.
    Grishagin, V., Israfilov, R., Sergeyev, Y.D.: Comparative efficiency of dimensionality reduction schemes in global optimization. AIP Conf. Proc. 1776, 060011-1–060011-4 (2016).
  23. 23.
    Gergel, V., Goryachih, A.: Multidimensional global optimization using numerical estimations of minimized function derivatives. Optim. Methods Softw. (2019).
  24. 24.
    Sergeyev, Y.D., Kvasov, D.E.: Deterministic Global Optimization: An Introduction to the Diagonal Approach. Springer, New York (2017). CrossRefzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Software and SupercomputingLobachevsky State UniversityNizhni NovgorodRussia

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