A modified orthogonal-descent algorithm for finding the zero of a complex function

Щепакин, М.Б.; Shubenkova, I. A.

doi:10.1007/bf01125866

Cited by 1 publication

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“…Although, it should be noted that, by using subgradient norms and variable parameter λ k , more meaningful methods of orthogonal subgradient descent can be provided. For example, by using the space dilation operator and the operator (39), it is possible to ensure the space transformation so that direction of movement in Y k coincides with the shortest vector to the convex shell of the accumulated subgradients. This direction of movement is used in ε-subgradient methods and is considered to be a good replacement for the Newtonian direction.…”

Section: Orthogonal Subgradient Descent Methods With a Classical Feje...mentioning

confidence: 99%

“…However, it does not always give good results in terms of the accuracy of solving the problem (5) by functional. This is evidenced by the results of a number of numerical experiments with both the orthogonal descent method and its modification, presented in [38,39]. Despite the small size of the problems (n ∼ 10) and the fact that they are not very robust, the accuracy of their solution by functional is often…”

Section: Theorem 1 ([9]mentioning

confidence: 99%

“…We will discuss the selection of the λ parameter and its meaning a little later. The operator (39) we will call a orthogonalizing one-rank operator. The following statements hold for it.…”

Section: Orthogonalizing One-rank Operatormentioning

confidence: 99%

“…3. Using the transformation (39), one can try to obtain computationally stable methods for solving systems of linear equations within the framework of the orthogonalization methods, which have proven to be numerically unstable [44,45]. At the same time, in addition to the transformation itself, there are a number of mechanisms both at the level of B form and H form for ensuring stable recalculation of the distances to hyperplanes in the transformed space of arguments.…”

mentioning

confidence: 99%

“…The operator (39) and such a step-by-step cone construction strategy leads to a number of variable metric methods, which are based on the localization of the extrema set in transformed space by an orthogonal cone defined by previously calculated subgradients. At the same time, a solution of the problem of limiting the number of stored subgradients is automatically provided.…”

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confidence: 99%

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

Semenov,

Stetsyuk,

Stovba

et al. 2024

Mathematics

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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.

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Section: Orthogonal Subgradient Descent Methods With a Classical Feje...mentioning

confidence: 99%

Section: Theorem 1 ([9]mentioning

confidence: 99%

“…We will discuss the selection of the λ parameter and its meaning a little later. The operator (39) we will call a orthogonalizing one-rank operator. The following statements hold for it.…”

Section: Orthogonalizing One-rank Operatormentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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