v2009.01.01 - Convex Optimization

164 CHAPTER 2. CONVEX GEOMETRY
that follows from (335) and leads to an equally peculiar halfspace-description
m∑
K ∗ = {y ∈ R m | y j A j ≽ 0} (343)
The summation inequality with respect to the positive semidefinite cone is
known as a linear matrix inequality. [51] [126] [226] [315]
When the A j matrices are linearly independent, function g(y) = ∆ ∑ y j A j
on R m is a linear bijection. Inverse image of the positive semidefinite cone
under g(y) must therefore have dimension m . In that circumstance, the
dual cone interior is nonempty
m∑
int K ∗ = {y ∈ R m | y j A j ≻ 0} (344)
having boundary
∂K ∗ = {y ∈ R m |
j=1
j=1
m∑
y j A j ≽ 0,
j=1
m∑
y j A j ⊁ 0} (345)
j=1
2.13.6 Dual of pointed polyhedral cone
In a subspace of R n , now we consider a pointed polyhedral cone K given in
terms of its extreme directions Γ i arranged columnar in
X = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (253)
The extremes theorem (2.8.1.1.1) provides the vertex-description of a
pointed polyhedral cone in terms of its finite number of extreme directions
and its lone vertex at the origin:
2.13.6.0.1 Definition. Pointed polyhedral cone, vertex-description.
Given pointed polyhedral cone K in a subspace of R n , denoting its i th extreme
direction by Γ i ∈ R n arranged in a matrix X as in (253), then that cone may
be described: (78) (confer (170) (265))
K = { [0 X ]aζ | a T 1 = 1, a ≽ 0, ζ ≥ 0 }
{
= Xaζ | a T 1 ≤ 1, a ≽ 0, ζ ≥ 0 }
{ } (346)
= Xb | b ≽ 0 ⊆ R
n
that is simply a conic hull (like (94)) of a finite number N of directions.
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