Open AccessA new approach to the Farkas theorem of the alternative
Author(s) -
Yu. G. Evtushenko,
Ю. Г. Евтушенко,
А. А. Тretyakov,
А А Третьяков,
Е. Е. Тыртышников,
Е. Е. Тыртышников
Publication year - 2019
Publication title -
doklady akademii nauk. rossijskaâ akademiâ nauk
Language(s) - English
Resource type - Journals
ISSN - 0869-5652
DOI - 10.31857/s0869-56524856655-658
Subject(s) - mathematical proof , assertion , mathematics , cone (formal languages) , vector space , analytic proof , bounded function , algebra over a field , discrete mathematics , calculus (dental) , pure mathematics , computer science , mathematical analysis , algorithm , geometry , medicine , dentistry , programming language
The classical Farkas theorem of the alternative is considered, which is widely used in various areas of mathematics and has numerous proofs and formulations. An entirely new elementary proof of this theorem is proposed. It is based on the consideration of a functional that, under Farkas’ condition, is bounded below on the whole space and attains a minimum. The assertion of Farkas’ theorem that a vector belongs to a cone is equivalent to the fact that the gradient of this functional is zero at the minimizer.