v2009.01.01 - Convex Optimization

6.10. POSTSCRIPT 493
When D is an EDM [223,2]
N(D) ⊂ N(1 T ) = {z | 1 T z = 0} (1203)
Because [137,2] (E.0.1)
then
DD † 1 = 1
1 T D † D = 1 T (1204)
R(1) ⊂ R(D) (1205)
6.10 postscript
We provided an equality (1173) relating the convex cone of Euclidean distance
matrices to the convex cone of positive semidefinite matrices. Projection on
a positive semidefinite cone, constrained by an upper bound on rank, is easy
and well known; [107] simply, a matter of truncating a list of eigenvalues.
Projection on a positive semidefinite cone with such a rank constraint is, in
fact, a convex optimization problem. (7.1.4)
In the past, it was difficult to project on the EDM cone under a
constraint on rank or affine dimension. A surrogate method was to invoke
the Schoenberg criterion (817) and then project on a positive semidefinite
cone under a rank constraint bounding affine dimension from above. But a
solution acquired that way is necessarily suboptimal.
In7.3.3 we present a method for projecting directly on the EDM cone
under a constraint on rank or affine dimension.