v2009.01.01 - Convex Optimization

476 CHAPTER 6. CONE OF DISTANCE MATRICES
In fact, the smallest face that contains auxiliary matrix V of the PSD
cone S N + is the intersection with the geometric center subspace (1874) (1875);
F ( S N + ∋V ) = cone { V N υυ T VN
T | υ ∈ RN−1}
= S N c ∩ S N +
≡ {X ≽ 0 | 〈X , 11 T 〉 = 0} (1479)
(1146)
In isometrically isomorphic R N(N+1)/2
svec F ( S N + ∋V ) = cone T (1147)
related to S N c by
aff cone T = svec S N c (1148)
6.7.2 EDM criteria in 11 T
(confer6.5) Laurent specifies an elliptope trajectory condition for EDM cone
membership: [202,2.3]
D ∈ EDM N ⇔ [1 − e −αd ij
] ∈ EDM N ∀α > 0 (997)
From the parametrized elliptope E N t
D ∈ EDM N ⇔ ∃ t∈ R +
E∈ E N t
in6.6.2 we propose
}
D = t11 T − E (1149)
Chabrillac & Crouzeix [65,4] prove a different criterion they attribute
to Finsler (1937) [120]. We apply it to EDMs: for D ∈ S N h (945)
−V T N DV N ≻ 0 ⇔ ∃κ>0 −D + κ11 T ≻ 0
⇔
D ∈ EDM N with corresponding affine dimension r=N −1
(1150)
This Finsler criterion has geometric interpretation in terms of the
vectorization & projection already discussed in connection with (1140). With
reference to Figure 122, the offset 11 T is simply a direction orthogonal to
T in isomorphic R 3 . Intuitively, translation of −D in direction 11 T is like
orthogonal projection on T in so far as similar information can be obtained.