Farkas lemma
Given an matrix and an real row vector , both with real coefficients, one and only one of the following systems has a solution:
- 1.
and for some -column vector ;
- 2.
and for some -row vector .
Equivalently, one and only one of the following has a solution:
- 1.
, and for some -column vector ;
- 2.
and for some -row vector .
Remark. Here, means that every of is nonnegative, and similarly with the other expressions.