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Theorems of Alternative and Optimization

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Optimization and Applications (OPTIMA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12422))

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Abstract

A new elementary proof of Farkas’ theorem is proposed. The proof is based on the consideration of the always solvable problem of minimizing the residual of a system of linear equations/inequalities and the necessary and sufficient conditions for the minimum of this problem. Minimizing the residuals of an inconsistent system makes it easy to calculate the normal solution to a consistent system. The connection between Farkas’ theorem and linear and quadratic programming is shown. A new version of the theorem on alternatives is proposed.

This work was supported in part by the Russian Foundation for Basic Research, project no. 17-07-00510.

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Author information

Authors and Affiliations

  1. Dorodnicyn Computing Centre, FRC “Computer Science and Control” of RAS, Vavilov Street 40, 119333, Moscow, Russia

    Yuri Evtushenko & Alexander Golikov

  2. Moscow Institute of Physics and Technology (State Research University), 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia

    Yuri Evtushenko

Authors
  1. Yuri Evtushenko
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  2. Alexander Golikov
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Corresponding author

Correspondence to Alexander Golikov .

Editor information

Editors and Affiliations

  1. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Nicholas Olenev

  2. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Yuri Evtushenko

  3. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  4. Dorodnicyn Computing Centre, FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

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Evtushenko, Y., Golikov, A. (2020). Theorems of Alternative and Optimization. In: Olenev, N., Evtushenko, Y., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2020. Lecture Notes in Computer Science(), vol 12422. Springer, Cham. https://doi.org/10.1007/978-3-030-62867-3_7

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