v2009.01.01 - Convex Optimization

478 CHAPTER 6. CONE OF DISTANCE MATRICES
The ordinary dual EDM cone cannot, therefore, be pointed. (2.13.1.1)
When N = 1, the EDM cone is the point at the origin in R . Auxiliary
matrix V N is empty [ ∅ ] , and dual cone EDM ∗ is the real line.
When N = 2, the EDM cone is a nonnegative real line in isometrically
isomorphic R 3 ; there S 2 h is a real line containing the EDM cone. Dual cone
EDM 2∗ is the particular halfspace in R 3 whose boundary has inward-normal
EDM 2 . Diagonal matrices {δ(u)} in (1153) are represented by a hyperplane
through the origin {d | [ 0 1 0 ]d = 0} while the term cone{V N υυ T V T N }
is represented by the halfline T in Figure 122 belonging to the positive
semidefinite cone boundary. The dual EDM cone is formed by translating
the hyperplane along the negative semidefinite halfline −T ; the union of
each and every translation. (confer2.10.2.0.1)
When cardinality N exceeds 2, the dual EDM cone can no longer be
polyhedral simply because the EDM cone cannot. (2.13.1.1)
6.8.1.1 EDM cone and its dual in ambient S N
Consider the two convex cones
so
K ∆ 1 = S N h
K ∆ 2 =
⋂
{
A ∈ S N | 〈yy T , −A〉 ≥ 0 }
y∈N(1 T )
= { A ∈ S N | −z T V AV z ≥ 0 ∀zz T (≽ 0) } (1155)
= { A ∈ S N | −V AV ≽ 0 }
K 1 ∩ K 2 = EDM N (1156)
Dual cone K ∗ 1 = S N⊥
h ⊆ S N (65) is the subspace of diagonal matrices.
From (1153) via (284),
K ∗ 2 = − cone { V N υυ T V T N | υ ∈ R N−1} ⊂ S N (1157)
Gaffke & Mathar [123,5.3] observe that projection on K 1 and K 2 have
simple closed forms: Projection on subspace K 1 is easily performed by
symmetrization and zeroing the main diagonal or vice versa, while projection
of H ∈ S N on K 2 is
P K2 H = H − P S N
+
(V H V ) (1158)