v2009.01.01 - Convex Optimization

5.9. BRIDGE: CONVEX POLYHEDRA TO EDMS 409
In fact, any positive semidefinite matrix whose entries belong to {±1} is
a correlation matrix whose rank must be one: (As there are few equivalent
conditions for rank constraints, this statement is rather important as a device
for relaxing an integer, combinatorial, or Boolean problem.)
5.9.1.0.2 Theorem. Some elliptope vertices. [104,2.1.1]
For Y ∈ S n , y ∈ R n , and all i,j∈{1... n} (confer2.3.1.0.2)
Y ≽ 0, Y ij ∈ {±1} ⇔ Y = yy T , y i ∈ {±1} (983)
⋄
The elliptope for dimension n = 2 is a line segment in isometrically
isomorphic R n(n+1)/2 (Figure 107). Obviously, cone(E n )≠ S n + . The elliptope
for dimension n = 3 is realized in Figure 106.
5.9.1.0.3 Lemma. Hypersphere. [16,4] (confer bullet p.358)
Matrix A = [A ij ]∈ S N belongs to the elliptope in S N iff there exist N points p
on the boundary of a hypersphere in R rank A having radius 1 such that
‖p i − p j ‖ = √ 2 √ 1 − A ij , i,j=1... N (984)
⋄
There is a similar theorem for Euclidean distance matrices:
We derive matrix criteria for D to be an EDM, validating (817) using
simple geometry; distance to the polyhedron formed by the convex hull of a
list of points (68) in Euclidean space R n .
5.9.1.0.4 EDM assertion.
D is a Euclidean distance matrix if and only if D ∈ S N h and distances-square
from the origin
{‖p(y)‖ 2 = −y T V T NDV N y | y ∈ S − β} (985)
correspond to points p in some bounded convex polyhedron
P − α = {p(y) | y ∈ S − β} (986)
having N or fewer vertices embedded in an r-dimensional subspace A − α
of R n , where α ∈ A = aff P and where the domain of linear surjection p(y)
is the unit simplex S ⊂ R N−1
+ shifted such that its vertex at the origin is
translated to −β in R N−1 . When β = 0, then α = x 1 .
⋄