Statement of The Lemma
Let A be an m × n matrix and b an m-dimensional vector. Then, exactly one of the following two statements is true:
- There exists an x ∈ Rn such that Ax = b and x ≥ 0.
- There exists a y ∈ Rm such that ATy ≥ 0 and bTy < 0.
Here, the notation x ≥ 0 means that all components of the vector x are nonnegative.
There are a number of slightly different (but equivalent) formulations of the Lemma in the literature. The one given here is due to Gale, Kuhn and Tucker in 1951.
Read more about this topic: Farkas' Lemma
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