Combined methods for solving degenerate unconstrained optimization problems
UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems.  The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order de...
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| Datum: | 2024 |
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Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7395 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512665150947328 |
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| author | Zadachyn, Viktor Bebiya, Maxim Zadachyn, Viktor Bebiya, Maxim |
| author_facet | Zadachyn, Viktor Bebiya, Maxim Zadachyn, Viktor Bebiya, Maxim |
| author_sort | Zadachyn, Viktor |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-27T08:49:56Z |
| description | UDC 519.853.6 : 519.613.2
We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems.  The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives.  Our approach is based on the decomposition of $\mathbb{R}^n$ into the direct sum of the kernel of a Hessian matrix and its orthogonal complement.  The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement.  This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix.  In order to get the second-order method, Newton's method is combined with the steepest-descent method.  We study the efficiency of these methods and analyze their convergence rates.  We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case.  The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions. |
| doi_str_mv | 10.3842/umzh.v76i5.7395 |
| first_indexed | 2026-03-24T03:32:24Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7395 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:24Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-73952024-06-27T08:49:56Z Combined methods for solving degenerate unconstrained optimization problems Combined methods for solving degenerate unconstrained optimization problems Zadachyn, Viktor Bebiya, Maxim Zadachyn, Viktor Bebiya, Maxim unconditional optimization, degenerate minimum point, optimality conditions, Newton’s modified method UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems.  The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives.  Our approach is based on the decomposition of $\mathbb{R}^n$ into the direct sum of the kernel of a Hessian matrix and its orthogonal complement.  The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement.  This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix.  In order to get the second-order method, Newton's method is combined with the steepest-descent method.  We study the efficiency of these methods and analyze their convergence rates.  We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case.  The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions. УДК 519.853.6 : 519.613.2 Комбіновані методи розв'язування вироджених задач безумовної оптимізації Наведено конструктивні методи другого та четвертого порядку для розв'язування вироджених безумовних задач оптимізації. Метод четвертого порядку, який ми використовуємо, є комбінацією методу Ньютона та методу, що використовує похідні четвертого порядку. Наш підхід базується на зображенні $\mathbb{R}^n$ як прямої суми ядра матриці Гесса та її ортогонального доповнення. До ядра матриці Гесса застосовано метод четвертого порядку, а до ортогонального доповнення --- метод Ньютона. Цей метод є ефективним у випадку одновимірного ядра матриці Гесса. Для отримання методу другого порядку метод Ньютона поєднано з методом найкрутішого спуску.  Досліджено продуктивність цих методів та проаналізовано швидкість їх збіжності. Також запропоновано новий адаптивний комбінований квазіньютонівський метод (ACQNM), що використовує методи другого та четвертого порядку для виродженого випадку.  Ефективність ACQNM показано на прикладі деяких найбільш поширених тестових функцій.  Institute of Mathematics, NAS of Ukraine 2024-06-02 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7395 10.3842/umzh.v76i5.7395 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 5 (2024); 695 - 718 Український математичний журнал; Том 76 № 5 (2024); 695 - 718 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7395/9939 Copyright (c) 2024 Задачин Віктор, Максим Бебія |
| spellingShingle | Zadachyn, Viktor Bebiya, Maxim Zadachyn, Viktor Bebiya, Maxim Combined methods for solving degenerate unconstrained optimization problems |
| title | Combined methods for solving degenerate unconstrained optimization problems |
| title_alt | Combined methods for solving degenerate unconstrained optimization problems |
| title_full | Combined methods for solving degenerate unconstrained optimization problems |
| title_fullStr | Combined methods for solving degenerate unconstrained optimization problems |
| title_full_unstemmed | Combined methods for solving degenerate unconstrained optimization problems |
| title_short | Combined methods for solving degenerate unconstrained optimization problems |
| title_sort | combined methods for solving degenerate unconstrained optimization problems |
| topic_facet | unconditional optimization degenerate minimum point optimality conditions Newton’s modified method |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7395 |
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