Advertisement

A Class of Smooth Modification of Space-Filling Curves for Global Optimization Problems

  • Alexey GoryachihEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 197)

Abstract

This work presents a class of smooth modifications of space-filling curves applied to global optimization problems. These modifications make the approximations of the Peano curves (evolvents) differentiable in all points, and save the differentiability of the optimized function. To evaluate the proposed approach, some results of numerical experiments with the original and modified evolvents for solving global optimization problems are discussed.

Notes

Acknowledgements

This research was supported by the Russian Science Foundation, project No 16-11-10150 Novel efficient methods and software tools for time-consuming decision-making problems using supercomputers of superior performance.

References

  1. 1.
    Trn, A., Ilinskas, A.: Global Optimization. Lecture Notes in Computer Science 350. Springer, Heidelberg (1989)Google Scholar
  2. 2.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Heidelberg (1990)CrossRefzbMATHGoogle Scholar
  3. 3.
    Zhigljavsky, A.A.: Theory of Global Random Search. Kluwer Academic Publishers, Dordrecht (1991)CrossRefGoogle Scholar
  4. 4.
    Pintr, J.D.: Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications). Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  5. 5.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Locatelli, M., Schoen, F.: Global optimization: theory, algorithms, and applications. SIAM (2013)Google Scholar
  7. 7.
    Floudas, C.A., Pardalos, M.P.: Recent Advances in Global Optimization. Princeton University Press, Princeton (2016)Google Scholar
  8. 8.
    Chendes, T. (ed.).: Development in Reliable Computing. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  9. 9.
    Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Paulaviius, R., Ilinskas, A.: Simplicial Global Optimization. Springer Briefs in Optimization. Springer, New York (2014)CrossRefGoogle Scholar
  11. 11.
    Strongin, R.G.: Numerical Methods in Multi-extremal Problems (Information-Statistical Algorithms). Nauka, Moscow (1978). (in Russian)zbMATHGoogle Scholar
  12. 12.
    Strongin, R.G., Gergel, V.P., Grishagin, V.A., Barkalov, K.A.: Parallel Computations for Global Optimization Problems. Moscow State University, Moscow (2013). (in Russian)Google Scholar
  13. 13.
    Barkalov, K.A., Gergel, V.P.: Multilevel scheme of dimensionality reduction for parallel global search algorithms. In: Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, pp. 2111–2124 (2014)Google Scholar
  14. 14.
    Gergel, V., Grishagin, V., Israfilov, R.: Local tuning in nested scheme of global optimization. Procedia Comput. Sci. 51, 865–874 (2015)CrossRefGoogle Scholar
  15. 15.
    Gergel V.P., Grishagin V.A., Gergel A.V.: Adaptive nested optimization scheme for multidimensional global search. J. Glob. Optim., 1–17 (2015)Google Scholar
  16. 16.
    Gergel, V., Sidorov, S. .A: Two-level parallel global search algorithm for solution of computationally intensive multiextremal optimization problems. In: Malyshkin, V. (Ed.) PaCT 2015, LNCS, vol. 9251, pp. 505–515. Springer, Heidelberg (2015)Google Scholar
  17. 17.
    Lera, D., Sergeyev, Y.D.: Acceleration of univariate global optimization algorithms working with lipschitz functions and lipschitz first derivatives. SIAM J. Optim. 23(1), 508–529 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sergeyev, Y.D., Kvasov, D.E.: A deterministic global optimization using smooth diagonal auxiliary functions. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 99–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kvasov, D.E., Sergeyev, Y.D.: Deterministic approaches for solving practical black-box global optimization problems. Adv. Eng. Softw. 80, 58–66 (2015)CrossRefGoogle Scholar
  20. 20.
    Breiman, L., Cutler, A.: A deterministic algorithm for global optimization. Math. Program. 58(1–3), 179–199 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gergel, V.P.: A method of using derivatives in the minimization of multiextremum functions. Comput. Math. Math. Phys. 36(6), 729–742 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81(1), 127–146 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Grishagin, V.A.: Operating characteristics of some global search algorithms. Probl. Stoch. Search. 7, 198–206 (1978). (In Russian)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Lobachevsky State University of Nizhni NovgorodNizhni NovgorodRussia

Personalised recommendations