anti-cone
Let X be a real vector space, and Φ be a subspace of linear functionals
on X.
For any set S⊆X, its anti-cone S+, with respect to Φ, is the set
S+={ϕ∈Φ:ϕ(x)≥0, for all x∈S}. |
The anti-cone is also called the dual cone.
Usage
The anti-cone operation is generally applied to subsets of X
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 Φ in the definition may be taken to be any subspace
of the algebraic dual space X*.
The set Φ often needs to be restricted
to a subspace smaller than X*, or even
the continuous dual space X′,
in order to obtain
the nice closure and reflexivity properties below.
Basic properties
Property 1.
The anti-cone is a convex cone in Φ.
Proof.
If ϕ(x) is non-negative, then so is tϕ(x) for t>0. And if ϕ1(x),ϕ2(x)≥0, then clearly (1-t)ϕ1(x)+tϕ2(x)≥0 for 0≤t≤1. ∎
Property 2.
If K⊆X is a cone, then its anti-cone K+ may be equivalently characterized as:
K+={ϕ∈Φ:ϕ(x) over x∈K is bounded below}. |
Topological properties
Assumptions.
Assume that Φ separates points of X.
Let X have the weak topology generated by Φ,
and let Φ have the weak-* topology generated by X;
this makes X and Φ into Hausdorff topological vector spaces
.
Vectors x∈X will be identified with their images ˆx under the natural embedding of X in its double dual space.
The pairing (X,Φ) is sometimes called a dual pair; and (Φ,X), where X is identified with its image in the double dual, is also a dual pair.
Property 3.
S+ is weak-* closed.
Proof.
Let {ϕα}⊆Φ be a net converging to ϕ
in the weak-* topology.
By definition, ˆx(ϕα)=ϕα(x)≥0.
As the functional ˆx is continuous
in the weak-* topology,
we have ˆx(ϕα)→ˆx(ϕ)≥0.
Hence ϕ∈S+.
∎
Property 4.
ˉS+=S+.
Proof.
The inclusion ˉS+⊆S+ is obvious. And if ϕ(x)≥0 for all x∈S, then by continuity, this holds true for x∈ˉS too — so ˉS+⊇S+. ∎
Properties involving cone inclusion
Property 5 (Farkas’ lemma).
Let K⊆X be a weakly-closed convex cone. Then x∈K if and only if ϕ(x)≥0 for all ϕ∈K+.
Proof.
That ϕ(x)≥0 for ϕ∈K+ and x∈K is just the definition.
For the converse, we show that if x∈X∖K,
then there exists ϕ∈K+ such that ϕ(x)<0.
If K=∅, then the desired ϕ∈Φ=K+ exists because Φ can separate the points x and 0. If K≠∅, by the hyperplane separation theorem, there is a ϕ∈Φ such that ϕ(x)<infy∈Kϕ(y). This ϕ will automatically be in K+ by Property 2. The zero vector is the weak limit of ty, as t↘0, for any vector y. Thus 0∈K, and we conclude with infy∈Kϕ(y)≤0. ∎
Property 6.
K++=ˉK for any convex cone K. (The anti-cone operation on K+ is to be taken with respect to X.)
Proof.
We work with ˉK, which is a weakly-closed convex cone. By Property 5, x∈ˉK if and only if ϕ(x)≥0 for all ϕ∈ˉK+=K+. But by definition of the second anti-cone, ˆx∈(K+)+ if and only if ϕ(x)=ˆx(ϕ)≥0 for all ϕ∈K+. ∎
Property 7.
Let K and L be convex cones in X, with K weakly closed. Then K+⊆L+ if and only if K⊇L.
Proof.
K+⊆L+⟹K=ˉK=K++⊇L++=ˉL⊇L⟹K+⊆L+.∎ |
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 |