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International Conference on Optimization and Applications

OPTIMA 2018: Optimization and Applications pp 49–63Cite as

Subgradient Method with Polyak’s Step in Transformed Space

Part of the Communications in Computer and Information Science book series (CCIS,volume 974)

Abstract

We consider two subgradient methods (methods A and B) for finding the minimum point of a convex function for the known optimal value of the function. Method A is a subgradient method, which uses the Polyak’s step in the original space of variables. Method B is a subgradient method in the transformed space of variables, which uses Polyak’s step in the transformed space. For both methods a proof of the convergence of finding the minimum point with a given accuracy by the value of the function was performed. Examples of ravine convex (smooth and non-smooth) functions are given, for which convergence of method A is slow. It is shown that with a suitable choice of the space transformation matrix method B can be significantly accelerated in comparison with method A for ravine convex functions.

Keywords

  • Subgradient method
  • Polyak’s step
  • Space transformation

Supported by Volkswagen Foundation (grant No. 90 306).

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Author information

Authors and Affiliations

  1. V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, 40 Glushkov ave., Kyiv, 03187, Ukraine

    Petro Stetsyuk & Viktor Stovba

  2. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37 Prosp. Peremohy, Kyiv, 03056, Ukraine

    Zhanna Chernousova

Authors
  1. Petro Stetsyuk
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  2. Viktor Stovba
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  3. Zhanna Chernousova
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Corresponding author

Correspondence to Viktor Stovba .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Prof. Yury Evtushenko

  2. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  3. Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  4. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

    Prof. Yury Kochetov

  5. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Vlasta Malkova

  6. FRC CSC RAS, Dorodnitsyn Computing Centre, Moscow, Russia

    Dr. Mikhail Posypkin

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Stetsyuk, P., Stovba, V., Chernousova, Z. (2019). Subgradient Method with Polyak’s Step in Transformed Space. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol 974. Springer, Cham. https://doi.org/10.1007/978-3-030-10934-9_4

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  • DOI: https://doi.org/10.1007/978-3-030-10934-9_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10933-2

  • Online ISBN: 978-3-030-10934-9

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