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Two Proofs of the Farkas-Minkowski Theorem by Tandem Method and by Use of an Orthogonal Complement

Author

Listed:
  • Takao Fujimoto

    (University of Kelaniya, Kelaniya)

  • B.B.Upeksha P. Perera

    (University of Kelaniya, Kelaniya)

  • Giorgio Giorgi

    (Department of Economics and Management, University of Pavia)

Abstract

This note presents two proofs of the Farkas-Minkowski theorem. The first one is analytical, and this does not presuppose the closedness of a finitely generated cone. We do not employ separation theorems either. Even the concept of linear independence or invertibility of matrices is not necessary. Our new device consists in proving the Farkas-Minkowski theorem and the closedness of a finitely generated cone at the same time based upon mathematical induction. We make use of a distance minimization problem with an equality constraint. The second proof is algebraic, and a mixture of Gale’s and Ben-Israel’s methods. Our proof based on the orthogonal complement seems easy to understand in terms of geometrical images.

Suggested Citation

  • Takao Fujimoto & B.B.Upeksha P. Perera & Giorgio Giorgi, 2016. "Two Proofs of the Farkas-Minkowski Theorem by Tandem Method and by Use of an Orthogonal Complement," DEM Working Papers Series 129, University of Pavia, Department of Economics and Management.
    • Handle: RePEc:pav:demwpp:demwp0129
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    File URL: http://dem-web.unipv.it/web/docs/dipeco/quad/ps/RePEc/pav/demwpp/DEMWP0129.pdf
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    References listed on IDEAS

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    4. Michele Conforti & Marco Di Summa & Giacomo Zambelli, 2007. "Minimally Infeasible Set-Partitioning Problems with Balanced Constraints," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 497-507, August.
    5. Filippini, Carlo & Filippini, Luigi, 1982. "Two Theorems on Joint Production," Economic Journal, Royal Economic Society, vol. 92(366), pages 386-390, June.
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