v2009.01.01 - Convex Optimization

466 CHAPTER 6. CONE OF DISTANCE MATRICES
and where {δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i , i=1... N} are extreme
directions of some pointed polyhedral cone K ⊂ S N h and extreme directions
of EDM N . Invertibility of (1118)
−V DV 1 = −V ∑ N (
λ
2 i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qi
T
i=1
∑
= N λ i q i qi
T
i=1
)
V
1
2
(1119)
implies linear independence of those extreme directions.
biorthogonal expansion is expressed
dvec D = Y Y † dvec D = Y λ ( )
−V DV 1 2
Then the
(1120)
where
Y ∆ = [ dvec ( δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i
)
, i = 1... N
]
∈ R N(N−1)/2×N (1121)
When D belongs to the EDM cone in the subspace of symmetric hollow
matrices, unique coordinates Y † dvecD for this biorthogonal expansion
must be the nonnegative eigenvalues λ of −V DV 1 . This means D
2
simultaneously belongs to the EDM cone and to the pointed polyhedral cone
dvec K = cone(Y ).
6.5.3.3 Open question
Result (1115) is analogous to that for the positive semidefinite cone (204),
although the question remains open whether all faces of EDM N (whose
dimension is less than dimension of the cone) are exposed like they are for
the positive semidefinite cone. 6.8 (2.9.2.3) [301]
6.6 Correspondence to PSD cone S N−1
+
Hayden, Wells, Liu, & Tarazaga [159,2] assert one-to-one correspondence
of EDMs with positive semidefinite matrices in the symmetric subspace.
Because rank(V DV )≤N−1 (5.7.1.1), that PSD cone corresponding to
6.8 Elementary example of a face not exposed is given in Figure 27 and Figure 36.