v2009.01.01 - Convex Optimization

490 CHAPTER 6. CONE OF DISTANCE MATRICES
Because 〈 {δ(u) | u∈ R N }, D 〉 ≥ 0 ⇔ D ∈ S N h , we can restrict observation
to the symmetric hollow subspace without loss of generality. Then for D ∈ S N h
〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ { −V N υυ T V T N | υ ∈ R N−1} ⇔ D ∈ EDM N (1198)
this discretized membership relation becomes (1195); identical to the
Schoenberg criterion.
Hitherto a correspondence between the EDM cone and a face of a PSD
cone, the Schoenberg criterion is now accurately interpreted as a discretized
membership relation between the EDM cone and its ordinary dual.
6.8.2 Ambient S N h
When instead we consider the ambient space of symmetric hollow matrices
(1154), then still we find the EDM cone is not self-dual for N >2. The
simplest way to prove this is as follows:
Given a set of generators G = {Γ} (1117) for the pointed closed convex
EDM cone, the discretized membership theorem in2.13.4.2.1 asserts that
members of the dual EDM cone in the ambient space of symmetric hollow
matrices can be discerned via discretized membership relation:
EDM N∗ ∩ S N h
∆
= {D ∗ ∈ S N h | 〈Γ , D ∗ 〉 ≥ 0 ∀ Γ ∈ G(EDM N )}
(1199)
By comparison
= {D ∗ ∈ S N h | 〈δ(zz T )1 T + 1δ(zz T ) T − 2zz T , D ∗ 〉 ≥ 0 ∀z∈ N(1 T )}
= {D ∗ ∈ S N h | 〈1δ(zz T ) T − zz T , D ∗ 〉 ≥ 0 ∀z∈ N(1 T )}
EDM N = {D ∈ S N h | 〈−zz T , D〉 ≥ 0 ∀z∈ N(1 T )} (1200)
the term δ(zz T ) T D ∗ 1 foils any hope of self-duality in ambient S N h .