Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1439310.31857/S0869-56524856655-658Research ArticleA new approach to the Farkas theorem of the alternativeEvtushenkoYu. G.<p>Academician of the Russian Academy of Sciences</p>evt@ccas.ruTret’yakovA. A.tret@ap.siedlce.plTyrtyshnikovE. E.<p>Academician of the Russian Academy of Sciences</p>eugene.tyrtyshnikov@gmail.comDorodnicyn Computing Centre, Federal Research Center "Informatics and Management" of the Russian Academy of SciencesLomonosov Moscow State UniversitySystem Research Institute of the Polish Academy of SciencesSiedlce University of Natural Sciences and HumanitiesMarchuk Institute of Numerical Mathematics of the Russian Academy of Sciences2405201948566556582506201925062019Copyright © 2019, Russian academy of sciences2019<p class="a"><span lang="EN-US">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.</span></p>Farkas’ theorem of the alternativeFarkas conditionWeyl’s theoremFourier-Motzkin elimination methodтеорема Фаркаша об альтернативеусловие Фаркашатеорема Вейляметод Фурье-Моцкина[Артамонов В. А., Латышев В. Н. Линейная алгебра и выпуклая геометрия. М.: Факториал Пресс, 2004.][Карманов В. Г. Математическое программирование. М.: Наука, 1986.][Тыртышников Е. Е. Основы алгебры. М.: Физматлит, 2017.][Bartl D. A Short Algebraic Proof of the Farkas Lemma // SIAM J. Optim. 2008. V. 19. № 1. P. 234-239.][Broyden C. G. A Simple Algebraic Proof of Farkas’s Lemma and Related Theorems // Optim. Methods and Software. 1998. V. 8. № 3/4. P. 185-199.][Chandru V., Lassez J. L. Qualitative Theorem Proving in Linear Constraints. In: Verification: Theory and Practice. B., Heidelberg: Springer, 2003. P. 395-406.][Dax A. Classroom Note: An Elementary Proof of Farkas’ Lemma // SIAM Rev. 1997. V. 39. № 3. P. 503-507.][Голиков А. И., Евтушенко Ю. Г. Теоремы об альтернативах и их применение в численных методах //ЖВМиМФ. 2003. Т. 43. № 3. С. 354-375.][Jaćimoviс́ M. Farkas’ Lemma of Alternative // The Teaching Math. 2011. V. 25. № 27. P. 77-86.][Marjanović M. M. An Iterative Method for Solving Polynomial Equations. In: Topology and Its Appl. Budva, 1972. P. 170-172.][Roos C., Terlaky T. S. Note on a Paper of Broyden //Operations Res. Lett. 1999. V. 25. № 4. P. 183-186.][Ziegler G. M. Lectures on Polytopes. B.: Springer-Verlag, 1995.]