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

Volodymyr Semenov, Petro Stetsyuk, Viktor Stovba and José Manuel Velarde Cantú ()
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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

Mathematics, 2024, vol. 12, issue 10, 1-26

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.

Keywords: convex programming; one-rank linear transformations; Fejer methods; Polyak step size; subgradients (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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