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Piecewise Linear Bounding Functions for Univariate Global Optimization

  • Oleg Khamisov
  • Mikhail Posypkin
  • Alexander UsovEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)

Abstract

The paper addresses the problem of constructing lower and upper bounding functions for univariate functions. This problem is of a crucial importance in global optimization where such bounds are used by deterministic methods to reduce the search area. It should be noted that bounding functions are expected to be relatively easy to construct and manipulate with. We propose to use piecewise linear estimators for bounding univariate functions. The rules proposed in the paper enable an automated synthesis of lower and upper bounds from the function’s expression in an algebraic form. Numerical examples presented in the paper demonstrate the high accuracy of the proposed bounds.

Keywords

Univariate global optimization Piecewise linear functions Estimators Deterministic methods 

References

  1. 1.
    Baritompa, W.: Accelerations for a variety of global optimization methods. J. Global Optim. 4(1), 37–45 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bompadre, A., Mitsos, A.: Convergence rate of McCormick relaxations. J. Global Optim. 52(1), 1–28 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Breiman, L., Cutler, A.: A deterministic algorithm for global optimization. Math. Program. 58(1–3), 179–199 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Casado, L.G., MartÍnez, J.A., GarcÍa, I., Sergeyev, Y.D.: New interval analysis support functions using gradient information in a global minimization algorithm. J. Global Optim. 25(4), 345–362 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ershov, A., Khamisov, O.V.: Automatic global optimization. Diskretnyi Analiz i Issledovanie Operatsii 11(2), 45–68 (2004)MathSciNetGoogle Scholar
  6. 6.
    Evtushenko, Y.G.: A numerical method of search for the global extremum of functions (scan on a nonuniform net). Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 11(6), 1390–1403 (1971)MathSciNetGoogle Scholar
  7. 7.
    Evtushenko, Y., Posypkin, M.: A deterministic approach to global box-constrained optimization. Optimization Letters 7(4), 819–829 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Floudas, C., Gounaris, C.: A review of recent advances in global optimization. J. Global Optim. 45(1), 3–38 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gergel, V., Grishagin, V., Israfilov, R.: Local tuning in nested scheme of global optimization. Procedia Comput. Sci. 51, 865–874 (2015)CrossRefGoogle Scholar
  10. 10.
    Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis: Revised and Expanded. CRC Press, New York (2003)Google Scholar
  11. 11.
    Horst, R., Pardalos, P.M.: Handbook of Global Optimization, vol. 2. Springer Science & Business Media, Dordrecht (2013)zbMATHGoogle Scholar
  12. 12.
    Khajavirad, A., Sahinidis, N.: Convex envelopes of products of convex and component-wise concave functions. J. Global Optim. 52(3), 391–409 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Khamisov, O.: Explicit univariate global optimization with piecewise linear support functions, proc. DOOR 2016, CEUR-WS.org, vol. 1623, pp. 218–255. http://ceur-ws.org/Vol-1623/papermp19.pdf
  14. 14.
    Khamisov, O.: Optimization with quadratic support functions in nonconvex smooth optimization, aIP Conference Proceedings 1776, 050010 (2016).  https://doi.org/10.1063/1.4965331
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pardalos, P.M., Rosen, J.: Reduction of nonlinear integer separable programming problems. Int. J. Comput. Math. 24(1), 55–64 (1988)CrossRefGoogle Scholar
  17. 17.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer, New York (2014). 10.1007/978-1-4614-9093-7CrossRefGoogle Scholar
  18. 18.
    Pijavskij, S.: An algorithm for finding the global extremum of function. Optimal Decisions 2, 13–24 (1967)Google Scholar
  19. 19.
    Pintér, J.D.: Global Optimization in Action: Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, vol. 6. Springer Science & Business Media, New York (2013)Google Scholar
  20. 20.
    Ratz, D.: An optimized interval slope arithmetic and its application. Inst. für Angewandte Mathematik (1996)Google Scholar
  21. 21.
    Ratz, D.: A nonsmooth global optimization technique using slopes: the one-dimensional case. J. Global Optim. 14(4), 365–393 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rosen, J.B., Pardalos, P.M.: Global minimization of large-scale constrained concave quadratic problems by separable programming. Math. Program. 34(2), 163–174 (1986)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sergeyev, Y.D.: An information global optimization algorithm with local tuning. SIAM J. Optim. 5(4), 858–870 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81(1), 127–146 (1998)MathSciNetCrossRefGoogle Scholar
  25. 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. Optim. Theory Appl. 171(1), 186–208 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shubert, B.O.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9(3), 379–388 (1972)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Strekalovsky, A.S.: Global optimality conditions for nonconvex optimization. J. Global Optim. 12(4), 415–434 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms, vol. 45. Springer Science & Business Media, New York (2013)zbMATHGoogle Scholar
  29. 29.
    Vinkó, T., Lagouanelle, J.L., Csendes, T.: A new inclusion function for optimization: Kite-the one dimensional case. J. Global Optim. 30(4), 435–456 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of SciencesIrkutskRussia
  2. 2.Dorodnicyn Computing Centre, FRC CSC RASMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyInstitutskiy Pereulok9 DolgoprudnyRussia
  4. 4.Institute for Information Transmission Problems RASMoscowRussia

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