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Automatic Convexity Deduction for Efficient Function’s Range Bounding

Author

Listed:
  • Mikhail Posypkin

    (Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilova 44-2, 119333 Moscow, Russia)

  • Oleg Khamisov

    (Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences, Lermontov St., 130, 664033 Irkutsk, Russia)

Abstract

Reliable bounding of a function’s range is essential for deterministic global optimization, approximation, locating roots of nonlinear equations, and several other computational mathematics areas. Despite years of extensive research in this direction, there is still room for improvement. The traditional and compelling approach to this problem is interval analysis. We show that accounting convexity/concavity can significantly tighten the bounds computed by interval analysis. To make our approach applicable to a broad range of functions, we also develop the techniques for handling nondifferentiable composite functions. Traditional ways to ensure the convexity fail in such cases. Experimental evaluation showed the remarkable potential of the proposed methods.

Suggested Citation

  • Mikhail Posypkin & Oleg Khamisov, 2021. "Automatic Convexity Deduction for Efficient Function’s Range Bounding," Mathematics, MDPI, vol. 9(2), pages 1-16, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:134-:d:477948
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    References listed on IDEAS

    as
    1. Strekalovsky, Alexander S., 2015. "On local search in d.c. optimization problems," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 73-83.
    2. Lera, Daniela & Posypkin, Mikhail & Sergeyev, Yaroslav D., 2021. "Space-filling curves for numerical approximation and visualization of solutions to systems of nonlinear inequalities with applications in robotics," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    3. Daniela Lera & Yaroslav D. Sergeyev, 2018. "GOSH: derivative-free global optimization using multi-dimensional space-filling curves," Journal of Global Optimization, Springer, vol. 71(1), pages 193-211, May.
    4. A. Žilinskas, 1978. "Optimization of One‐Dimensional Multimodal Functions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(3), pages 367-375, November.
    5. Robert Fourer & Chandrakant Maheshwari & Arnold Neumaier & Dominique Orban & Hermann Schichl, 2010. "Convexity and Concavity Detection in Computational Graphs: Tree Walks for Convexity Assessment," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 26-43, February.
    6. Orhan Arıkan & Regina S. Burachik & C. Yalçın Kaya, 2020. "Steklov regularization and trajectory methods for univariate global optimization," Journal of Global Optimization, Springer, vol. 76(1), pages 91-120, January.
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    Cited by:

    1. Mikhail Posypkin & Andrey Gorshenin & Vladimir Titarev, 2022. "Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”," Mathematics, MDPI, vol. 10(13), pages 1-8, June.

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