Section
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Mathematics
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Title
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Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema
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Author(-s)
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Galkin O.E.a,
Galkina S.Yu.a
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Affiliations
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National Research University Higher School of Economics, Nizhni Novgoroda
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Abstract
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For real-valued functions $f$, defined on subsets of real linear spaces, the notions of extreme subarguments, extreme epiarguments, natural convex $\check{f}$ and natural concave $\hat{f}$ envelopes are introduced. It is shown that for any strictly convex function $g$, any point of the global maximum of the function $f+g$ is an extreme subargument for the function $f$. A similar result is obtained for functions of the form $f/v + g$. Based on these results, a method is proposed, that facilitates the search for global extrema of functions in some cases. It is proved that under certain conditions the functions $f/v+g$ and $\hat{f}/v+g$ have the same global maximum and the same points of the global maximum. Necessary and sufficient conditions for the naturalness of the convex envelope of function are given. A sufficient condition for the invariance of values of the concave envelope $\hat{f}$ during narrowing the domain of $f$ is established. Extreme sub- and epiarguments for continuous nowhere differentiable Gray-Takagi function $K(x)$ of Kobayashi on the segment $[0;1]$ are found. Moreover, the global extrema of the function $K(x)/\cos{x}$ and the global maximum of the function $K(x)-\sqrt{x(1-x)}$ on $[0;1]$ are calculated. The article is provided with examples and graphic illustrations.
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Keywords
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nondifferentiable optimization, extreme subarguments (subabscissae) and epiarguments (epiabscissae) of function, natural convex and concave envelopes of function, Gray Takagi function of Kobayashi
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UDC
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517.518.244, 519.6
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MSC
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26A27, 26A30, 26B25, 49M30, 90C26
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DOI
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10.20537/vm190402
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Received
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16 September 2019
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Language
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Russian
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Citation
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Galkin O.E., Galkina S.Yu. Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 483-500.
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