Properties and Application of Incomplete Orthogonalization in the Directions of Gradient Difference in Optimization Methods : научное издание

Krutikov Vladimir; Tovbis Elena; Gutova Svetlana; Rozhnov Ivan; Kazakovtsev Lev

Дополнительные ссылки: eLibrary

База данных: Каталог библиотеки СФУ

Библиографическое описание: Properties and Application of Incomplete Orthogonalization in the Directions of Gradient Difference in Optimization Methods : научное издание / V. V. Krutikov [et al.]. - Текст : непосредственный // Mathematics. - 2025. - Т. 13, № 24. - P. 4036. - ISSN 2227-7390, DOI 10.3390/math13244036.

Аннотация: This paper considers the problem of unconstrained minimization of smooth functions. Despite the high efficiency of quasi-Newton methods such as BFGS, their performance degrades in ill-conditioned problems with unstable or rapidly varying Hessians—for example, in functions with curved ravine structures. This necessitates alternative approaches that rely not on second-derivative approximations but on the topological properties of level surfaces. As a new methodological framework, we propose using a procedure of incomplete orthogonalization in the directions of gradient differences, implemented through the iterative least-squares method (ILSM). Two new methods are constructed based on this approach: a gradient method with the ILSM metric (HY_g) and a modification of the Hestenes–Stiefel conjugate gradient method with the same metric (HY_XS). Both methods are shown to have linear convergence on strongly convex functions and finite convergence on quadratic functions. A numerical experiment was conducted on a set of test functions. The results show that the proposed methods significantly outperform BFGS (2 times for HY_g and 3.5 times for HY_XS in terms of iterations number) when solving ill-posed problems with varying Hessians or complex level topologies, while providing comparable or better performance even in high-dimensional problems. This confirms the potential of using topology-based metrics alongside classical quasi-Newton strategies.

Год издания: 2025

Авторы: Krutikov Vladimir , Tovbis Elena , Gutova Svetlana , Rozhnov Ivan , Kazakovtsev Lev

Издательство: MDPI, Basel

Источник: Mathematics

Выпуск: Т. 13, № 24

Номера страниц: 4036

Держатель оригинала документа: Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia, Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia, Laboratory “Hybrid Methods of Modeling and Optimization in Complex Systems”, Siberian Federal University, 79 Svobodny Prospekt, Krasnoyarsk 660041, Russia

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Properties and Application of Incomplete Orthogonalization in the Directions of Gradient Difference in Optimization Methods : научное издание
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