Farkas' lemma

Farkas's lemma is a result in mathematics stating that a vector is either in a given convex cone or that there exists a (hyper)plane separating the vector from the cone—there are no other possibilities. It was originally proven by the Hungarian mathematician Gyula Farkas.^[1] It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming.

Farkas's lemma is an example of a theorem of alternatives: a theorem stating that of two systems, one or the other has a solution, but not both nor none.

Statement of the lemma

Let $\mathbf {A}$ be a real $m\times n$ matrix and $\mathbf {b}$ an $m$ -dimensional real vector. Then, exactly one of the following two statements is true:

There exists an $\mathbf {x} \in \mathbb {R} ^{n}$ such that $\mathbf {Ax} =\mathbf {b}$ and $\mathbf {x} \geq 0$ .
There exists a $\mathbf {y} \in \mathbb {R} ^{m}$ such that $\mathbf {y} ^{\mathtt {T}}\mathbf {A} \geq 0$ and $\mathbf {y} ^{\mathtt {T}}\mathbf {b} <0$ .

Here, the notation $\mathbf {x} \geq 0$ means that all components of the vector $\mathbf {x}$ are nonnegative. If we write $C(\mathbf {A} )$ for the cone generated by the columns of $\mathbf {A}$ , then the vector $\mathbf {x}$ proves that $\mathbf {b}$ lies in $C(\mathbf {A} )$ while the vector $\mathbf {y}$ gives a linear functional that separates $\mathbf {b}$ from $C(\mathbf {A} )$ .

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 & Tucker (1951).

Geometric interpretation

Let a₁, …, a_n ∈ R^m denote the columns of A. In terms of these vectors, Farkas's lemma states that exactly one of the following two statements is true:

There exist coefficients x₁, …, x_n ∈ R, x₁, …, x_n ≥ 0, such that b = x₁a₁ + ··· + x_na_n.
There exists a vector y ∈ R^m such that a_i ^T · y ≥ 0 for i = 1, …, n and b · y < 0.

The vectors x₁a₁ + ··· + x_na_n with nonnegative coefficients constitute the convex cone of the set {a₁, …, a_n} so the first statement says that b is in this cone.

The second statement says that there exists a vector y such that the angle of y with the vectors a_i is at most 90° while the angle of y with the vector b is more than 90°. The hyperplane normal to this vector has the vectors a_i on one side and the vector b on the other side. Hence, this hyperplane separates the vectors in the cone of {a₁, …, a_n} and the vector b.

For example, let n,m=2 and a₁ = (1,0)^T and a₂ = (1,1)^T. The convex cone spanned by a₁ and a₂ can be seen as a wedge-shaped slice of the first quadrant in the x-y plane. Now, suppose b = (0,1). Certainly, b is not in the convex cone a₁x₁+a₂x₂. Hence, there must be a separating hyperplane. Let y = (1,−1)^T. We can see that a₁ · y = 1, a₂ · y = 0, and b · y = −1. Hence, the hyperplane with normal y indeed separates the convex cone a₁x₁+a₂x₂ from b.

Farkas's lemma can thus be interpreted geometrically as follows: Given a convex cone and a vector, either the vector is in the cone or there is a hyperplane separating the vector from the cone, but not both.

Further implications

Farkas's lemma can be varied to many further theorems of alternative by simple modifications, such as Gordan's theorem: Either $Ax < 0$ has a solution x, or $A^T y = 0$ has a nonzero solution y with y ≥ 0.

Common applications of Farkas' lemma include proving the strong and weak duality theorem associated with linear programming, game theory at a basic level and the Kuhn–Tucker constraints. An extension of Farkas' lemma can be used to analyze the strong duality conditions for and construct the dual of a semidefinite program. 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 $Ax$ ≤ $b$ is unsolvable then $y^T A = 0$ , $y^T b = -1$ , $y$ ≥ $0$ has a solution.^[2] (Note that $y^T A$ is a combination of the left hand sides, $y^T b$ 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.)

Notes

↑ Farkas (1894, 1902)
↑ Boyd, Stephen P.; Vandenberghe, Lieven (2004), "Section 5.8.3", Convex Optimization (pdf), Cambridge University Press, ISBN 978-0-521-83378-3, retrieved October 15, 2011

References

Berkovitz, Leonard D. (2001), Convexity and Optimization in $\mathbb {R} ^{n}$ , New York: John Wiley & Sons, ISBN 978-0-471-35281-5 .
Farkas, Gyula (1894), "A Fourier-féle mechanikai elv alkamazásai", Mathematikai és Természettudományi Értesítő, 12: 457–472 . reference from Schrijver's Combinatorial Optimization textbook
Farkas, Julius (Gyula) (1902), "Über die Theorie der Einfachen Ungleichungen", Journal für die Reine und Angewandte Mathematik, 124 (124): 1–27, doi:10.1515/crll.1902.124.1 .
Rockafellar, R. T. (1979), Convex Analysis, Princeton University Press, p. 200
Gale, David; Kuhn, Harold; Tucker, Albert W. (1951), "Linear Programming and the Theory of Games - Chapter XII", in Koopmans, Activity Analysis of Production and Allocation (PDF), Wiley . See Lemma 1 on page 318.
Kutateladze, S. (2010), "The Farkas Lemma Revisited" (PDF), Siberian Mathematical Journal, 51 (1): 78–87, doi:10.1007/s11202-010-0010-y.

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