New Approach to Farkas’ Theorem of the Alternative


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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.

作者简介

Yu. Evtushenko

Dorodnitsyn Computing Center, Federal Research Center “Computer Science and Control” of the Russian Academy
of Sciences,; Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University; Moscow Institute of Physics and Technology (State University); Moscow Aviation Institute (National Research University)

编辑信件的主要联系方式.
Email: Yuri-evtushenko@yandex.ru
俄罗斯联邦, Moscow, 119333; Moscow, 119991; Dolgoprudnyi, Moscow oblast, 141700; Moscow, 125080

A. Tret’yakov

Dorodnitsyn Computing Center, Federal Research Center “Computer Science and Control” of the Russian Academy
of Sciences,; System Research Institute, Polish Academy of Sciences; Faculty of Sciences, Siedlce University

编辑信件的主要联系方式.
Email: tret@uph.edu.pl
俄罗斯联邦, Moscow, 119333; Warsaw, 01-447; Siedlce, 08-110

E. Tyrtyshnikov

Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University; Faculty of Sciences, Siedlce University; Institute of Numerical Mathematics, Russian Academy
of Sciences

编辑信件的主要联系方式.
Email: eugene.tyrtyshnikov@gmail.com
俄罗斯联邦, Moscow, 119991; Siedlce, 08-110; Moscow, 119333

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