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Subgradient Method with Polyak’s Step in Transformed Space

  • Petro Stetsyuk
  • Viktor StovbaEmail author
  • Zhanna Chernousova
Conference paper
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 

References

  1. 1.
    Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9(3), 14–29 (1969)CrossRefGoogle Scholar
  2. 2.
    Agmon, S.: The relaxation method for linear inequalities. Can. J. Math. 6(3), 382–392 (1954)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Motzkin, T., Schoenberg, I.J.: The relaxation method for linear inequalities. Can. J. Math. 6(3), 393–404 (1954)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eremin, I.I.: Generalization of the Motzkin-Agmon relaxational method. Uspekhi Mat. Nauk. 20(2), 183–187 (1965)MathSciNetGoogle Scholar
  5. 5.
    Sergienko, I.V., Stetsyuk, P.I.: On N.Z. Shor’s three scientific ideas. Cybern. Syst. Anal. 48(1), 2–16 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Shor, N.Z.: Minimization Methods for Non-Differentiable Functions. SSCM, vol. 3. Springer, Berlin (1985).  https://doi.org/10.1007/978-3-642-82118-9CrossRefzbMATHGoogle Scholar
  7. 7.
    Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers, Boston (1998)CrossRefGoogle Scholar
  8. 8.
    Stetsyuk, P.I.: Methods of Ellipsoids and r-Algorithms. Eureka, Chisinau (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.V.M. Glushkov Institute of Cybernetics of NAS of UkraineKyivUkraine
  2. 2.National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”KyivUkraine

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