kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-1-209-219kaz29-2515Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESНЕОБХОДИМЫЕ И ДОСТАТОЧНЫЕ УСЛОВИЯ ЭКСТРЕМУМА ЦЕЛЕВОЙ ФУНКЦИИNECESSARY AND SUFFICIENT CONDITIONS FOR THE GLOBAL EXTREMUM OF OBJECTIVE FUNCTIONShttps://orcid.org/0009-0009-4471-1625ОтаровХ. Т.OtarovKh. T.

к.ф.-м.н.

г. Актобе

Cand. Phys.-Math. Sc.

Aktobe

khassenotar@mail.ruhttps://orcid.org/0000-0003-3611-9620ТуткушеваЖ. С.TutkushevaZh. S.

PhD

г. Актобе

PhD

Aktobe

zhailan_k@mail.ruАктюбинский региональный университет им. К. ЖубановаКазахстанAktobe Regional University named after K. ZhubanovKazakhstan202629032026231209219Copyright © Отаров Х.Т., Туткушева Ж.С., 20262026Отаров Х.Т., Туткушева Ж.С.Otarov K.T., Tutkusheva Z.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2515

Глобальная оптимизация многоэкстремальных функций многих переменных – важная задача развития различных направлений науки. Актуальность этой задачи подтверждается тем, что необходимость поиска глобального экстремума постоянно возникает как в теоретических исследованиях, так и на практике. В данной работе предлагается новый подход к проблеме поиска глобального минимума многоэкстремальной функции многих переменных. В опубликованной ранее работе одного из авторов данной статьи путем преобразования целевой функции была построена специальная функция, которая была названа «вспомогательной функцией». При этом были изучены важные свойства этой функции (неотрицательность, равномерная непрерывность, дифференцируемость, монотонность и другие), благодаря которым переход от целевой функции к вспомогательной обеспечивает экономию и гарантию сходимости метода поиска глобального минимума. В предлагаемой статье строго сформулированы и доказаны необходимые и достаточные условия глобального минимума целевой функции, и, как следствие, задача нахождения глобального минимума многоэкстремальной функции многих переменных сводится к поиску «наибольшего нуля» выпуклой функции одной переменной. При этом доказано, что глобальный минимум целевой функции совпадает с точной верхней гранью нулей вспомогательной функции. А для определения «наибольшего нуля» вспомогательной функции с высокой точностью можно рационально использовать известные численные методы.

Global optimization of multi-extremal, multi-variable functions is an important problem for the development of various areas of science. Its relevance is that the need to search for global extremum of functions constantly arises both in theoretical research and in practice. In this work, a new method for determining the global minimum of a multi-extremal, multi-variable function is proposed. In a previously published work by one of the authors of this article, a special function called the "auxiliary function" was constructed by transforming the objective function, and its important properties (non-negativity, uniform discontinuity, differentiability, monotonicity, etc.) were studied. In the presented article, necessary and sufficient conditions for the global minimum of the objective function are rigorously formulated and proven. As a result, the problem of finding the global minimum of a multi-extremal and multi-variable function was reduced to the problem of determining the "greatest zero" of a convex function of one variable: it was proved that the global minimum of the objective function is equal to the exact upper bound of the zeros of the auxiliary function. And the problem of rational application of known numerical methods to determine the "greatest zero" of the auxiliary function with high accuracy was considered.

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The authors declare that there are no conflicts of interest present.