vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"vestifm-248Research ArticleМАТЕМАТИКАMATHEMATICSО ПСЕВДОЛИПШИЦЕВОСТИ МНОЖЕСТВА РЕШЕНИЙ ПАРАМЕТРИЧЕСКИХ ЗАДАЧ ОПТИМИЗАЦИИPSEUDO-LIPSCHITZIAN CONTINUITY OF SOLUTION MAPPINGS IN PARAMETRICAL OPTIMIZATION PROBLEMSМинченкоЛ. И.MinchenkoL. I.

доктор физико-мате­матических наук, профессор, профессор кафедры информатики

D. Sc. (Physics and Mathematics), Professor, Professor of the Department of Informatics

inform@bsuir.byБережновД. Е.BerezhnovD. E.

ассистент, кафе­дра информатики

Assistant of the Department of Informatics

daniilberezhnov@gmail.comБелорусский государственный университет информатики и радиоэлектроникиBelarusian State University of Informatics and Radioelectronics201706082017023643Copyright © Минченко Л.И., Бережнов Д.Е., 20172017Минченко Л.И., Бережнов Д.Е.Minchenko L.I., Berezhnov D.E.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/248

Исследование свойств множества решений параметрических задач оптимизации представляет собой достаточно актуальную проблему. Значительные усилия направлены, в частности, на поиск условий различных типов обобщенной липшицевости множества решений, в частности условий их устойчивости (calmness) и псевдолипшицевости (Aubin property) [1]. Новый интересный подход к исследованию устойчивости множества решений предложен в работе М. Кановас и др. [2] в случае параметрической задачи линейного программирования и распространен Д. Клатте и Б. Куммером [3] на существенно более широкий круг задач. В данном подходе устойчивость множества решений связывается с устойчивостью некоторой ассоциированной системы, представляющей ограничение множества уровня целевой функции на множестве допустимых точек задачи. В настоящей статье предлагается расширить применение подхода [3] на исследование псевдолипшицевости множества решений; представлены некоторые достаточные условия псевдолипшицевости множества решений, а также обобщение леммы Хоффмана. 

The study of the properties of solution mappings in parametrical optimization problems represents an urgent problem. Particularly, considerable efforts are directed to finding the conditions of different types of generalized Lipschitzian continuity of solution mappings, namely their calmness and pseudo-Lipschitzian continuity (also referred to as the Aubin property) [1]. A new interesting approach to investigating the calmness of solution mappings has recently been proposed by Canovas et al. [2] for parametrical linear programming problems and applied to a much wider range of problems by Klatte and Kummer [3]. In this approach, the calmness of solution mappings is related to the calmness of an associated system representing a constraint on the level set of the objective function on the domain of the problem. In our note, we propose to expand the use of the approach [3] for investigating the pseudo-Lipschitzian continuity of solution mappings. Several sufficient conditions for the pseudo-Lipschitzian continuity of solution mappings, as well as the generalization of the Hoffman lemma are presented.

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