v2009.01.01 - Convex Optimization

6.1. DEFINING EDM CONE 447
6.1 Defining EDM cone
We invoke a popular matrix criterion to illustrate correspondence between the
EDM and PSD cones belonging to the ambient space of symmetric matrices:
{
−V DV ∈ S
N
D ∈ EDM N
+
⇔
(822)
D ∈ S N h
where V ∈ S N is the geometric centering matrix (B.4). The set of all EDMs
of dimension N ×N forms a closed convex cone EDM N because any pair of
EDMs satisfies the definition of a convex cone (157); videlicet, for each and
every ζ 1 , ζ 2 ≥ 0 (A.3.1.0.2)
ζ 1 V D 1 V + ζ 2 V D 2 V ≽ 0
ζ 1 D 1 + ζ 2 D 2 ∈ S N h
⇐ V D 1V ≽ 0, V D 2 V ≽ 0
D 1 ∈ S N h , D 2 ∈ S N h
(1081)
and convex cones are invariant to inverse linear transformation [266, p.22].
6.1.0.0.1 Definition. Cone of Euclidean distance matrices.
In the subspace of symmetric matrices, the set of all Euclidean distance
matrices forms a unique immutable pointed closed convex cone called the
EDM cone: for N > 0
EDM N = ∆ { }
D ∈ S N h | −V DV ∈ S N +
= ⋂ {
D ∈ S N | 〈zz T , −D〉≥0, δ(D)=0 } (1082)
z∈N(1 T )
The EDM cone in isomorphic R N(N+1)/2 [sic] is the intersection of an infinite
number (when N >2) of halfspaces about the origin and a finite number
of hyperplanes through the origin in vectorized variable D = [d ij ] . Hence
EDM N has empty interior with respect to S N because it is confined to the
symmetric hollow subspace S N h . The EDM cone relative interior comprises
rel int EDM N = ⋂ {
D ∈ S N | 〈zz T , −D〉>0, δ(D)=0 }
z∈N(1 T )
= { D ∈ EDM N | rank(V DV ) = N −1 } (1083)
while its relative boundary comprises
rel ∂EDM N = { D ∈ EDM N | 〈zz T , −D〉 = 0 for some z ∈ N(1 T ) }
= { D ∈ EDM N | rank(V DV ) < N −1 } (1084)
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