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Eurasian Mathematical Journal, 2023, Volume 14, Number 1, Pages 8–15
DOI: https://doi.org/10.32523/2077-9879-2023-14-1-08-15
(Mi emj458)
 

This article is cited in 1 scientific paper (total in 1 paper)

On the Lagrange multiplier rule for minimizing sequences

A. V. Arutyunov, S. E. Zhukovskiy

V.A. Trapeznikov Institute of Control Sciences of RAS, 65 Profsoyuznaya St, 117997 Moscow, Russia
Full-text PDF (351 kB) Citations (1)
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Abstract: In the paper, an optimization problem with equality-type constraints is studied. It is assumed that the minimizing function and the functions defining the constraints are Frechet differentiable, the set of the admissible points is nonempty and the minimizing function is bounded below on the set of admissible points. Under these assumptions we obtain an estimate of the derivative of the Lagrange function. Moreover, we prove the existence of a minimizing sequence $\{x^n\}$ and a sequence of unit Lagrange multipliers $\{\lambda^n\}$ such that the sequence of the values of derivative of the Lagrange function at the point $(x^n, \lambda^n)$ tends zero. This result is a generalization of the known assertion stating that for a bounded below differentiable function $f$ there exists a minimizing sequence $\{x^n\}$ such that the values of the derivative $f'(x^n)$ tend to zero. As an auxiliary tool, there was introduced and studied the property of the directional covering for mappings between normed spaces. There were obtained sufficient conditions of directional covering for Frechet differentiable mappings.
Keywords and phrases: constraint optimization, Lagrange multiplier rule, optimality condition, minimizing sequence, Caristi-like condition.
Funding agency Grant number
Russian Science Foundation 20-11-20131
22-11-00042
Theorem 2.1 and Lemma 3.2 were obtained by the first author under the financial support of the Russian Science Foundation (project no. 20-11-20131). Theorem 2.2 and Lemma 3.1 were obtained by the second author under the financial support of the Russian Science Foundation (project no. 22-11-00042).
Received: 09.12.2022
Bibliographic databases:
Document Type: Article
MSC: 49K27
Language: English
Citation: A. V. Arutyunov, S. E. Zhukovskiy, “On the Lagrange multiplier rule for minimizing sequences”, Eurasian Math. J., 14:1 (2023), 8–15
Citation in format AMSBIB
\Bibitem{AruZhu23}
\by A.~V.~Arutyunov, S.~E.~Zhukovskiy
\paper On the Lagrange multiplier rule for minimizing sequences
\jour Eurasian Math. J.
\yr 2023
\vol 14
\issue 1
\pages 8--15
\mathnet{http://mi.mathnet.ru/emj458}
\crossref{https://doi.org/10.32523/2077-9879-2023-14-1-08-15}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4575856}
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  • This publication is cited in the following 1 articles:
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