v2009.01.01 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1
+ 469
6.6.0.0.2 Example. ⎡ Extreme ⎤ rays versus rays on the boundary.
0 1 4
The EDM D = ⎣ 1 0 1 ⎦ is an extreme direction of EDM 3 where
[ ]
4 1 0
1
u = in (1123). Because −VN 2
TDV N has eigenvalues {0, 5} , the ray
whose direction is D also lies on[ the relative ] boundary of EDM 3 .
0 1
In exception, EDM D = κ , for any particular κ > 0, is an
1 0
extreme direction of EDM 2 but −VN TDV N has only one eigenvalue: {κ}.
Because EDM 2 is a ray whose relative boundary (2.6.1.3.1) is the origin,
this conventional boundary does not include D which belongs to the relative
interior in this dimension. (2.7.0.0.1)
6.6.1 Gram-form correspondence to S N−1
+
With respect to D(G)=δ(G)1 T + 1δ(G) T − 2G (810) the linear Gram-form
EDM operator, results in5.6.1 provide [1,2.6]
EDM N = D ( V(EDM N ) ) ≡ D ( )
V N S N−1
+ VN
T
(1128)
V N S N−1
+ VN T ≡ V ( D ( ))
V N S N−1
+ VN T = V(EDM N ) = ∆ −V EDM N V 1 = 2 SN c ∩ S N +
(1129)
a one-to-one correspondence between EDM N and S N−1
+ .
6.6.2 EDM cone by elliptope
(confer5.10.1) Defining the elliptope parametrized by scalar t>0
E N t = S N + ∩ {Φ∈ S N | δ(Φ)=t1} (1002)
then following Alfakih [8] we have
EDM N = cone{11 T − E N 1 } = {t(11 T − E N 1 ) | t ≥ 0} (1130)
Identification E N = E N 1 equates the standard elliptope (5.9.1.0.1,
Figure 106) to our parametrized elliptope.