v2009.01.01 - Convex Optimization

172 CHAPTER 2. CONVEX GEOMETRY
K ∗ ∩ aff K must be pointed if rel int K is logically assumed nonempty with
respect to aff K .
Conversely, suppose full-rank skinny-or-square matrix
[ ]
X †T =
∆ Γ ∗ 1 Γ ∗ 2 · · · Γ ∗ N ∈ R n×N (361)
comprises the extreme directions {Γ ∗ i } ⊂ aff K of the dual-cone intersection
with the affine hull of K . 2.67 From the discretized membership theorem and
(284) we get a partial dual to (347); id est, assuming x∈aff cone X
{ }
x ∈ K ⇔ γ ∗T x ≥ 0 for all γ ∗ ∈ Γ ∗ i , i=1... N ⊂ ∂K ∗ ∩ aff K (362)
⇔ X † x ≽ 0 (363)
that leads to a partial halfspace-description,
K = { x∈aff cone X | X † x ≽ 0 } (364)
For γ ∗ =X †T e i , any x =Xa , and for all i we have e T i X † Xa = e T i a ≥ 0
only when a ≽ 0. Hence x∈ K .
When X is full-rank, then the unique biorthogonal expansion of x ∈ K
becomes (357)
x = XX † x =
N∑
i=1
Γ i Γ ∗T
i x (365)
whose coordinates Γ ∗T
i x must be nonnegative because K is assumed pointed
closed and convex. Whenever X is full-rank, so is its pseudoinverse X † .
(E) In the present case, the columns of X †T are linearly independent and
generators of the dual cone K ∗ ∩ aff K ; hence, the columns constitute its
extreme directions. (2.10) That section of the dual cone is itself a polyhedral
cone (by (259) or the cone intersection theorem,2.7.2.1.1) having the same
number of extreme directions as K .
2.13.8.2 x ∈ aff K
The extreme directions of K and K ∗ ∩ aff K have a distinct relationship;
because X † X = I , then for i,j = 1... N , Γ T i Γ ∗ i = 1, while for i ≠ j ,
2.67 When closed convex cone K has empty interior, K ∗ has no extreme directions.