Lemma of Farkas

The lemma of Farkas is a result of Mathématiques used in particular in order to give a proof to the theorem of theorem of Karush-Kuhn-Tucker in nonlinear Programmation. The result of this lemma is according to: if $A$ is a matrix and $b$ a Vecteur, then one and only one of the following linear systems have a solution:

$Ax = B \,$ for $x \ geq 0$ ;
or $A^Ty \ geq 0$ for $y \,$ with $b^Ty < 0 \,$ .

This lemma was historically shown for the first time by Gyula Farkas in 1902 ( Über die Theorie der Einfachen Ungleichungen , Journal für die Reine und Angewandte Mathematik 124, p1-27) with a different formulation. The preceding formulation is due to Albert William Tucker in the years 1950.

It is an example of “theorem of the alternative”, which states that one or the other of two equations has a solution but not both simultaneously.

Literature

Julius Farkas. " Über die Theory der Einfachen Ungleichungen". Newspaper für die Queen und Angewandte Mathematik 124. p1-27

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