anti-cone
Let be a real vector space, and be a subspace of linear functionals
on .
For any set , its anti-cone , with respect to , is the set
The anti-cone is also called the dual cone.
Usage
The anti-cone operation is generally applied to subsets of
that are themselves
cones.
Recall that a cone in a real vector space generalize the notion of
linear inequalities in a finite number of real variables.
The dual cone provides a natural way to transfer such
inequalities
in the original vector space
to its dual space
.
The concept is useful in the theory of
duality.
The set in the definition may be taken to be any subspace
of the algebraic dual space .
The set often needs to be restricted
to a subspace smaller than , or even
the continuous dual space ,
in order to obtain
the nice closure and reflexivity properties below.
Basic properties
Property 1.
The anti-cone is a convex cone in .
Proof.
If is non-negative, then so is for . And if , then clearly for . ∎
Property 2.
If is a cone, then its anti-cone may be equivalently characterized as:
Topological properties
Assumptions.
Assume that separates points of .
Let have the weak topology generated by ,
and let have the weak-* topology generated by ;
this makes and into Hausdorff topological vector spaces
.
Vectors will be identified with their images under the natural embedding of in its double dual space.
The pairing is sometimes called a dual pair; and , where is identified with its image in the double dual, is also a dual pair.
Property 3.
is weak-* closed.
Proof.
Let be a net converging to
in the weak-* topology.
By definition, .
As the functional is continuous
in the weak-* topology,
we have .
Hence .
∎
Property 4.
.
Proof.
The inclusion is obvious. And if for all , then by continuity, this holds true for too — so . ∎
Properties involving cone inclusion
Property 5 (Farkas’ lemma).
Let be a weakly-closed convex cone. Then if and only if for all .
Proof.
That for and is just the definition.
For the converse, we show that if ,
then there exists such that .
If , then the desired exists because can separate the points and . If , by the hyperplane separation theorem, there is a such that . This will automatically be in by Property 2. The zero vector is the weak limit of , as , for any vector . Thus , and we conclude with . ∎
Property 6.
for any convex cone . (The anti-cone operation on is to be taken with respect to .)
Proof.
We work with , which is a weakly-closed convex cone. By Property 5, if and only if for all . But by definition of the second anti-cone, if and only if for all . ∎
Property 7.
Let and be convex cones in , with weakly closed. Then if and only if .
Proof.
References
-
1
B. D. Craven and J. J. Kohila.
“Generalizations
of Farkas’ Theorem
.” SIAM Journal on Mathematical Analysis. Vol. 8, No. 6, November 1977.
- 2 David G. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, 1969.
Title | anti-cone |
---|---|
Canonical name | Anticone |
Date of creation | 2013-03-22 17:20:48 |
Last modified on | 2013-03-22 17:20:48 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 8 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 46A03 |
Classification | msc 46A20 |
Synonym | anticone |
Synonym | dual cone |
Related topic | GeneralizedFarkasLemma |