v2009.01.01 - Convex Optimization

174 CHAPTER 2. CONVEX GEOMETRY
These results are summarized for a pointed polyhedral cone, having
linearly independent generators, and its ordinary dual:
2.13.9.2 Simplicial case
Cone Table 1 K K ∗
vertex-description X X †T , ±X ⊥
halfspace-description X † , X ⊥T X T
When a convex cone is simplicial (2.12.3), Cone Table 1 simplifies because
then aff coneX = R n : For square X and assuming simplicial K such that
rank(X ∈ R n×N ) = N ∆ = dim aff K = n (373)
we have
Cone Table S K K ∗
vertex-description X X †T
halfspace-description X † X T
For example, vertex-description (372) simplifies to
K ∗ = {X †T b | b ≽ 0} ⊂ R n (374)
Now, because dim R(X)= dim R(X †T ) , (E) the dual cone K ∗ is simplicial
whenever K is.
2.13.9.3 Cone membership relations in a subspace
It is obvious by definition (270) of the ordinary dual cone K ∗ in ambient vector
space R that its determination instead in subspace M⊆ R is identical to
its intersection with M ; id est, assuming closed convex cone K ⊆ M and
K ∗ ⊆ R
(K ∗ were ambient M) ≡ (K ∗ in ambient R) ∩ M (375)
because
{
y ∈ M | 〈y , x〉 ≥ 0 for all x ∈ K
}
=
{
y ∈ R | 〈y , x〉 ≥ 0 for all x ∈ K
}
∩ M
(376)