v2009.01.01 - Convex Optimization

422 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
Vertex-description of the dual spectral cone is, (284)
K ∗ λ = K∗ M + [
R
N
+
R −
] ∗
+ ∂H ∗ ⊆ R N+1
= {[ A T B T 1 −1 ] b | b ≽ 0 } =
From (1021) and (399) we get a halfspace-description:
{ }
[ÂT ˆBT 1 −1]
a | a ≽ 0
(1025)
K ∗ λ = {y ∈ R N+1 | (V T N [ÂT ˆBT ]) † V T N y ≽ 0} (1026)
This polyhedral dual spectral cone K ∗ λ is closed, convex, has nonempty
interior because K λ is pointed, but is not pointed because K λ has empty
interior.
5.11.2.3 Unordered eigenspectra
Spectral cones are not unique; eigenspectra ordering can be rendered benign
within a cone by presorting a vector of eigenvalues into nonincreasing
order. 5.50 Then things simplify: Conditions (1012) now specify eigenvalue
membership to the spectral cone
([ ]) [ ]
0 1
T R
N
λ
1 −EDM N +
= ∩ ∂H
R −
= {ζ ∈ R N+1 | Bζ ≽ 0, 1 T ζ = 0}
(1027)
where B is defined in (1019), and ∂H in (1017). From (398) we get a
vertex-description for a pointed spectral cone having empty interior:
([ ])
0 1
T {
λ
1 −EDM N = V N ( ˜BV
}
N ) † b | b ≽ 0
{[ ] } (1028)
I
=
−1 T b | b ≽ 0
5.50 Eigenspectra ordering (represented by a cone having monotone description such as
(1016)) becomes benign in (1237), for example, where projection of a given presorted vector
on the nonnegative orthant in a subspace is equivalent to its projection on the monotone
nonnegative cone in that same subspace; equivalence is a consequence of presorting.