# Farkas' lemma

### From Wikimization

m (→Extended Farkas' lemma) |
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From the definition of dual cone | From the definition of dual cone | ||

- | <math>\,\mathcal{K}^*\!=\{b~|~b^{\rm T}y\!\ | + | <math>\,\mathcal{K}^*\!=\{b~|~b^{\rm T}y\!\geq0~~\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> | ||

Line 27: | Line 27: | ||

<math>A^Ty\succeq0~\Leftrightarrow~b^Ty\geq0~~\forall~b\in\{A_{}x~|~x\succeq0\}</math> | <math>A^Ty\succeq0~\Leftrightarrow~b^Ty\geq0~~\forall~b\in\{A_{}x~|~x\succeq0\}</math> | ||

- | Given some <math>{\displaystyle b}</math> vector and <math>y\!\in | + | Given some <math>{\displaystyle b}</math> vector and <math>y\!\in\mathcal{K}~</math>, then <math>{\displaystyle b^Ty\!<0}</math> can only mean <math>b\notin\mathcal{K}^*</math>. |

An alternative system is therefore simply <math>b\in\mathcal{K}^*</math> | An alternative system is therefore simply <math>b\in\mathcal{K}^*</math> |

## Revision as of 21:19, 4 December 2011

**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.

## Contents |

## Proof

**(**Dattorro, ch.2.13**)** 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://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261361

# 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*.

### notes

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

(Sándor Zoltán Németh) Let be arbitrary. Then, by Moreau's theorem we have

and

Therefore,

In particular, for any we have Hence, Similarly, for any we have Hence, Therefore,