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This article is cited in 2 scientific papers (total in 2 papers)
MATHEMATICS
Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema
O. E. Galkin, S. Yu. Galkina National Research University Higher School of Economics, ul. Bolshaya Pecherskaya, 25/12, Nizhni Novgorod,
603155, Russia
Abstract:
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.
Keywords:
nondifferentiable optimization, extreme subarguments (subabscissae) and epiarguments (epiabscissae) of function, natural convex and concave envelopes of function, Gray Takagi function of Kobayashi.
Received: 16.09.2019
Citation:
O. E. Galkin, S. Yu. Galkina, “Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:4 (2019), 483–500
Linking options:
https://www.mathnet.ru/eng/vuu696 https://www.mathnet.ru/eng/vuu/v29/i4/p483
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