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One-Rank Linear Transformations and Fejer-Type Methods: An Overview

Author

Listed:
  • Volodymyr Semenov

    (Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 03022 Kyiv, Ukraine)

  • Petro Stetsyuk

    (Department of Nonsmooth Optimization Methods, V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, 03187 Kyiv, Ukraine)

  • Viktor Stovba

    (Department of Nonsmooth Optimization Methods, V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, 03187 Kyiv, Ukraine)

  • José Manuel Velarde Cantú

    (Department of Industrial Engineering, Technological Institute of Sonora (ITSON), Navojoa 85800, Sonora, Mexico)

Abstract

Subgradient methods are frequently used for optimization problems. However, subgradient techniques are characterized by slow convergence for minimizing ravine convex functions. To accelerate subgradient methods, special linear non-orthogonal transformations of the original space are used. This paper provides an overview of these transformations based on Shor’s original idea. Two one-rank linear transformations of Euclidean space are considered. These simple transformations form the basis of variable metric methods for convex minimization that have a natural geometric interpretation in the transformed space. Along with the space transformation, a search direction and a corresponding step size must be defined. Subgradient Fejer-type methods are analyzed to minimize convex functions, and Polyak step size is used for problems with a known optimal objective value. Convergence theorems are provided together with the results of numerical experiments. Directions for future research are discussed.

Suggested Citation

  • Volodymyr Semenov & Petro Stetsyuk & Viktor Stovba & José Manuel Velarde Cantú, 2024. "One-Rank Linear Transformations and Fejer-Type Methods: An Overview," Mathematics, MDPI, vol. 12(10), pages 1-26, May.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:10:p:1527-:d:1394346
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    References listed on IDEAS

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    1. Ivan V. Sergienko, 2012. "Methods of Optimization and Systems Analysis for Problems of Transcomputational Complexity," Springer Optimization and Its Applications, Springer, edition 127, number 978-1-4614-4211-0, August.
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