Farkas' lemma
From Wikimization
Farkas' lemma is a result used in the proof of the Karush-Kuhn-Tucker (KKT) theorem from nonlinear programming.
It states that if is a matrix and
a vector, then exactly one of the following two systems has a solution:
-
for some
such that
or in the alternative
-
for some
where the notation means that all components of the vector
are nonnegative.
The lemma was originally proved by Farkas in 1902. The above formulation is due to Albert W. Tucker in the 1950s.
It is an example of a theorem of the alternative; a theorem stating that of two systems, one or the other has a solution, but not both.
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Proof
(Dattorro, ch.2.13) Define a convex cone
whose dual cone is
From the definition of dual cone
we get
rather,
Given some vector and
, then
can only mean
.
An alternative system is therefore simply
and so the stated result follows.
Geometrical Interpretation
Farkas' lemma simply states that either vector belongs to convex cone
or it does not.
When , then there is a vector
normal to a hyperplane separating point
from cone
.
References
- Gyula Farkas, Über die Theorie der Einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, volume 124, pages 1–27, 1902.
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Extended Farkas' lemma
For any closed convex cone in the Hilbert space
(in particular we can have
, denote by
the polar cone of
Let be an arbitrary closed convex cone in
Then, the extended Farkas' lemma asserts that
Hence, denoting it follows that
Therefore, the cones and
are called mutually polar pair of cones.
notes
For definition of convex cone see in finite dimension see Convex cones, Wikimization.
For definition of polar cone see Moreau's theorem.
Proof of extended Farkas' lemma
(Sándor Zoltán Németh) Let be arbitrary. Then, by Moreau's theorem we have
and
Therefore,
In particular, for any we have
Hence,
Similarly, for any
we have
Hence,
Therefore,