v2009.01.01 - Convex Optimization

5.11. EDM INDEFINITENESS 421
So because of the hyperplane,
indicating K λ has empty interior. Defining
A ∆ =
⎡
⎢
⎣
e T 1 − e T 2
e T 2 − e T 3
.
e T N − eT N+1
we have the halfspace-description:
⎤
dim aff K λ = dim ∂H = N (1018)
⎥
⎦ ∈ RN×N+1 , B ∆ =
⎡
⎢
⎣
e T 1
e T2 .
e T N
−e T N+1
⎤
⎥
⎦ ∈ RN+1×N+1 (1019)
K λ = {ζ ∈ R N+1 | Aζ ≽ 0, Bζ ≽ 0, 1 T ζ = 0} (1020)
From this and (398) we get a vertex-description for a pointed spectral cone
having empty interior:
{ ([ ] ) †
}
Â
K λ = V N V N b | b ≽ 0
(1021)
ˆB
where V N ∈ R N+1×N , and where [sic]
ˆB = e T N ∈ R 1×N+1 (1022)
and
⎡
e T 1 − e T ⎤
2
 = ⎢
e T 2 − e T 3 ⎥
⎣ . ⎦ ∈ RN−1×N+1 (1023)
e T N−1 − eT N
hold [ ] those rows of A and B corresponding to conically independent rows in
A
V
B N .
Conditions (1012) can be equivalently restated in terms of a spectral cone
for Euclidean distance matrices:
⎧ ([ ]) [ ]
⎪⎨ 0 1
T
R
N
+
λ
∈ K
D ∈ EDM N ⇔ 1 −D
M ∩ ∩ ∂H
R − (1024)
⎪⎩
D ∈ S N h