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.
(Dattorro, ch.2.13) Define a convex cone
whose dual cone is
From the definition of dual cone we get
Given some vector and , then can only mean .
An alternative system is therefore simply and so the stated result follows.
Farkas' lemma simply states that either vector belongs to convex cone or it does not.
When , then there is a vector normal to a hyperplane separating point from cone .
- Gyula Farkas, Über die Theorie der Einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, volume 124, pages 1–27, 1902.
Extended Farkas' lemma
For any closed convex cone in the Hilbert space (in particular we can have , denote by the polar cone of
Let be an arbitrary closed convex cone in
Then, the extended Farkas' lemma asserts that
Hence, denoting it follows that
Therefore, the cones and are called mutually polar pair of cones.
For definition of convex cone see in finite dimension see Convex cones, Wikimization.
For definition of polar cone see Moreau's theorem.
Proof of extended Farkas' lemma
In particular, for any we have Hence, Similarly, for any we have Hence, Therefore,