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
Famous quotes containing the words statement of the, statement of and/or statement:
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)