v2009.01.01 - Convex Optimization

6.6. CORRESPONDENCE TO PSD CONE S N−1
+ 471
6.6.2.0.1 Expository. Define T E (11 T ) to be the tangent cone to the
elliptope E at point 11 T ; id est,
T E (11 T ) ∆ = {t(E − 11 T ) | t≥0} (1131)
The normal cone K ⊥ E (11T ) to the elliptope at 11 T is a closed convex cone
defined (E.10.3.2.1, Figure 150)
K ⊥ E (11 T ) ∆ = {B | 〈B , Φ − 11 T 〉 ≤ 0, Φ∈ E } (1132)
The polar cone of any set K is the closed convex cone (confer (270))
K ◦ ∆ = {B | 〈B , A〉≤0, for all A∈ K} (1133)
The normal cone is well known to be the polar of the tangent cone,
and vice versa; [173,A.5.2.4]
K ⊥ E (11 T ) = T E (11 T ) ◦ (1134)
K ⊥ E (11 T ) ◦ = T E (11 T ) (1135)
From Deza & Laurent [96, p.535] we have the EDM cone
EDM = −T E (11 T ) (1136)
The polar EDM cone is also expressible in terms of the elliptope. From
(1134) we have
EDM ◦ = −K ⊥ E (11 T ) (1137)
⋆
In5.10.1 we proposed the expression for EDM D
D = t11 T − E ∈ EDM N (1003)
where t∈ R + and E belongs to the parametrized elliptope E N t . We further
propose, for any particular t>0
Proof. Pending.
EDM N = cone{t11 T − E N t } (1138)
Relationship of the translated negated elliptope with the EDM cone is
illustrated in Figure 121. We speculate
EDM N = lim
t→∞
t11 T − E N t (1139)