Further Implications
Farkas's lemma can be varied to many further theorems of alternative by simple modifications, such as Gordan's theorem: Either has a solution x, or has a nonzero solution y with y ≥ 0.
Common applications of Farkas's lemma include proving the strong and weak duality theorem associated with linear programming, game theory at a basic level and the Kuhn-Tucker constraints. It is sufficient to prove the existence of the Kuhn-Tucker constraints using the Fredholm alternative but for the condition to be necessary, one must apply the Von Neumann equilibrium theorem to show the equations derived by Cauchy are not violated.
A particularly suggestive and easy-to-remember version is the following: if a set of inequalities has no solution, then a contradiction can be produced from it by linear combination with nonnegative coefficients. In formulas: if ≤ is unsolvable then, ≥ has a solution. (Note that is a combination of the left hand sides, a combination of the right hand side of the inequalities. Since the positive combination produces a zero vector on the left and a −1 on the right, the contradiction is apparent.)
Read more about this topic: Farkas' Lemma
Famous quotes containing the word implications:
“When it had long since outgrown his purely medical implications and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”
—Thomas Mann (1875–1955)
“Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the implications and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.”
—Susanne K. Langer (1895–1985)