Farkas' lemma
From Wikimization
Farkas' lemma is a result used in the proof of the Karush-Kuhn-Tucker (KKT) theorem from nonlinear programming.
It states that if is a matrix and a vector, then exactly one of the following two systems has a solution:
- for some such that
or in the alternative
- for some
where the notation means that all components of the vector are nonnegative.
The lemma was originally proved by Farkas in 1902. The above formulation is due to Albert W. Tucker in the 1950s.
It is an example of a theorem of the alternative; a theorem stating that of two systems, one or the other has a solution, but not both.
Proof
(Dattorro) Define a convex cone
whose dual cone is
From the definition of dual cone,
rather,
Given some vector and , then can only mean .
An alternative system is therefore simply and so the stated result follows.
References
- Gyula Farkas, Über die Theorie der Einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, volume 124, pages 1–27, 1902.
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