v2009.01.01 - Convex Optimization

484 CHAPTER 6. CONE OF DISTANCE MATRICES
6.8.1.3 Dual Euclidean distance matrix criterion
Conditions necessary for membership of a matrix D ∗ ∈ S N to the dual
EDM cone EDM N∗ may be derived from (1153): D ∗ ∈ EDM N∗ ⇒
D ∗ = δ(y) − V N AVN
T for some vector y and positive semidefinite matrix
A∈ S N−1
+ . This in turn implies δ(D ∗ 1)=δ(y) . Then, for D ∗ ∈ S N
where, for any symmetric matrix D ∗
D ∗ ∈ EDM N∗ ⇔ δ(D ∗ 1) − D ∗ ≽ 0 (1179)
δ(D ∗ 1) − D ∗ ∈ S N c (1180)
To show sufficiency of the matrix criterion in (1179), recall Gram-form
EDM operator
D(G) = δ(G)1 T + 1δ(G) T − 2G (810)
where Gram matrix G is positive semidefinite by definition, and recall the
self-adjointness property of the main-diagonal linear operator δ (A.1):
〈D , D ∗ 〉 = 〈 δ(G)1 T + 1δ(G) T − 2G , D ∗〉 = 〈G , δ(D ∗ 1) − D ∗ 〉 2 (829)
Assuming 〈G , δ(D ∗ 1) − D ∗ 〉≥ 0 (1386), then we have known membership
relation (2.13.2.0.1)
D ∗ ∈ EDM N∗ ⇔ 〈D , D ∗ 〉 ≥ 0 ∀D ∈ EDM N (1181)
Elegance of this matrix criterion (1179) for membership to the dual
EDM cone is the lack of any other assumptions except D ∗ be symmetric.
(Recall: Schoenberg criterion (817) for membership to the EDM cone requires
membership to the symmetric hollow subspace.)
Linear Gram-form EDM operator (810) has adjoint, for Y ∈ S N
Then we have: [78, p.111]
D T (Y ) ∆ = (δ(Y 1) − Y ) 2 (1182)
EDM N∗ = {Y ∈ S N | δ(Y 1) − Y ≽ 0} (1183)
the dual EDM cone expressed in terms of the adjoint operator. A dual EDM
cone determined this way is illustrated in Figure 127.