v2009.01.01 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1
+ 467
the EDM cone can only be S N−1
+ . [7,18.2.1] To clearly demonstrate this
correspondence, we invoke inner-product form EDM definition
D(Φ) ∆ =
[
0
δ(Φ)
]
1 T + 1 [ 0 δ(Φ) ] [ 0 0 T T
− 2
0 Φ
⇔
Φ ≽ 0
]
∈ EDM N
(922)
Then the EDM cone may be expressed
EDM N = { }
D(Φ) | Φ ∈ S N−1
+
(1122)
Hayden & Wells’ assertion can therefore be equivalently stated in terms of
an inner-product form EDM operator
D(S N−1
+ ) = EDM N (924)
V N (EDM N ) = S N−1
+ (925)
identity (925) holding because R(V N )= N(1 T ) (805), linear functions D(Φ)
and V N (D)= −VN TDV N (5.6.2.1) being mutually inverse.
In terms of affine dimension r , Hayden & Wells claim particular
correspondence between PSD and EDM cones:
r = N −1: Symmetric hollow matrices −D positive definite on N(1 T ) correspond
to points relatively interior to the EDM cone.
r < N −1: Symmetric hollow matrices −D positive semidefinite on N(1 T ) , where
−VN TDV N has at least one 0 eigenvalue, correspond to points on the
relative boundary of the EDM cone.
r = 1: Symmetric hollow nonnegative matrices rank-one on N(1 T ) correspond
to extreme directions (1117) of the EDM cone; id est, for some nonzero
vector u (A.3.1.0.7)
rankV T N DV N =1
D ∈ S N h ∩ R N×N
+
}
⇔
{
D ∈ EDM N
−V
T
D is an extreme direction ⇔ N DV N ≡ uu T
D ∈ S N h
(1123)