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A proof of the Farkas–Minkowski theorem by a tandem method
Author(s) -
Fujimoto Takao,
Perera B. B. Upeksha P.,
Giorgi Giorgio
Publication year - 2018
Publication title -
metroeconomica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.256
H-Index - 29
eISSN - 1467-999X
pISSN - 0026-1386
DOI - 10.1111/meca.12173
Subject(s) - mathematics , cone (formal languages) , minkowski space , constraint (computer aided design) , mathematical induction , minkowski's theorem , independence (probability theory) , pure mathematics , algebra over a field , calculus (dental) , discrete mathematics , algorithm , geometry , medicine , statistics , dentistry
This note presents a proof of the Farkas–Minkowski theorem. Our proof does not presuppose the closedness of a finitely generated cone, nor employs 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 minimization problem with an equality constraint, a method familiar to economics students.