Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema

O. E. Galkin ; S. Yu. Galkina

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 4, pp. 483-500

Résumé

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.

Détail

Keywords: nondifferentiable optimization, natural convex and concave envelopes of function, Gray Takagi function of Kobayashi.
Mots-clés : extreme subarguments (subabscissae) and epiarguments (epiabscissae) of function