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New Approach to Farkas’ Theorem of the Alternative

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Abstract

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.

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Authors and Affiliations

  1. Dorodnitsyn Computing Center, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences,, 119333, Moscow, Russia

    Yu. G. Evtushenko & A. A. Tret’yakov

  2. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991, Moscow, Russia

    Yu. G. Evtushenko &  E. E. Tyrtyshnikov

  3. Moscow Institute of Physics and Technology (State University), 141700, Dolgoprudnyi, Moscow oblast, Russia

    Yu. G. Evtushenko

  4. Moscow Aviation Institute (National Research University), 125080, Moscow, Russia

    Yu. G. Evtushenko

  5. System Research Institute, Polish Academy of Sciences, 01-447, Warsaw, Poland

    A. A. Tret’yakov

  6. Faculty of Sciences, Siedlce University, 08-110, Siedlce, Poland

    A. A. Tret’yakov &  E. E. Tyrtyshnikov

  7. Institute of Numerical Mathematics, Russian Academy of Sciences, 119333, Moscow, Russia

    E. E. Tyrtyshnikov

Authors
  1. Yu. G. Evtushenko
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  2. A. A. Tret’yakov
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  3. E. E. Tyrtyshnikov
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Corresponding authors

Correspondence to Yu. G. Evtushenko, A. A. Tret’yakov or E. E. Tyrtyshnikov.

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Translated by I. Ruzanova

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Evtushenko, Y.G., Tret’yakov, A.A. & Tyrtyshnikov, E.E. New Approach to Farkas’ Theorem of the Alternative. Dokl. Math. 99, 208–210 (2019). https://doi.org/10.1134/S1064562419020327

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  • DOI: https://doi.org/10.1134/S1064562419020327

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