# Farkas' lemma

### From Wikimization

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* <math>\quad\mathcal{K}^*=\{A_{}x~|~x\succeq0\}</math> | * <math>\quad\mathcal{K}^*=\{A_{}x~|~x\succeq0\}</math> | ||

- | From the definition of dual cone, | + | From the definition of dual cone |

+ | <math>\,\mathcal{K}^*\!=\{b~|~b^{\rm T}y\!\geq\!0~~\forall~y\!\in_{}\!\mathcal{K}\}</math> we get | ||

<math>y\in\mathcal{K}~\Leftrightarrow~b^Ty\geq0~~\forall~b\in\mathcal{K}^*</math> | <math>y\in\mathcal{K}~\Leftrightarrow~b^Ty\geq0~~\forall~b\in\mathcal{K}^*</math> |

## Revision as of 14:32, 12 November 2008

**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 we get

rather,

Given some vector and , then can only mean .

An alternative system is therefore simply and so the stated result follows.

## Geometrical Interpretation

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 .

## References

- Gyula Farkas, Über die Theorie der Einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, volume 124, pages 1–27, 1902.

http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D261364