v2009.01.01 - Convex Optimization

2.7. CONES 93
Then a pointed closed convex cone, by principle of separating hyperplane
(2.4.2.7), has a strictly supporting hyperplane at the origin. The simplest
and only bounded [324, p.75] convex cone K = {0} ⊆ R n is pointed by
convention, but has empty interior. Its relative boundary is the empty set
(154) while its relative interior is the point itself (11). The pointed convex
cone that is a halfline emanating from the origin in R n has the origin as
relative boundary while its relative interior is the halfline itself, excluding
the origin.
2.7.2.1.3 Theorem. Pointed cones. [48,3.3.15, exer.20]
A closed convex cone K ⊂ R n is pointed if and only if there exists a normal α
such that the set
C ∆ = {x ∈ K | 〈x,α〉=1} (162)
is closed, bounded, and K = cone C . Equivalently, K is pointed if and only
if there exists a vector β normal to a hyperplane strictly supporting K ;
id est, for some positive scalar ǫ
〈x, β〉 ≥ ǫ‖x‖ ∀x∈ K (163)
If closed convex cone K is not pointed, then it has no extreme point. 2.27
Yet a pointed closed convex cone has only one extreme point [37,3.3]:
the exposed point residing at the origin; its vertex. And from the cone
intersection theorem it follows that an intersection of convex cones is pointed
if at least one of the cones is.
2.7.2.2 Pointed closed convex cone and partial order
Relation ≼ is a partial order on some set if the relation possesses 2.28
reflexivity (x≼x)
antisymmetry (x≼z , z ≼x ⇒ x=z)
⋄
transitivity (x≼y , y ≼z ⇒ x≼z),
(x≼y , y ≺z ⇒ x≺z)
2.27 nor does it have extreme directions (2.8.1).
2.28 A set is totally ordered if it further obeys a comparability property of the relation
(a.k.a, trichotomy law): for each and every x and y from the set, x ≼ y or y ≼ x.