v2009.01.01 - Convex Optimization

358 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
D ∈ EDM N
⇔
{
−V DV ∈ S
N
+
D ∈ S N h
(822)
Of particular utility when D ∈ EDM N is the fact, (B.4.2 no.20) (794)
tr ( −V DV 1 2)
=
1
2N
∑
i,j
d ij = 1
2N vec(X)T (
∑
i,j
Φ ij ⊗ I
= tr(V GV ) , G ≽ 0
)
vec X
∑
= trG = N ‖x l ‖ 2 = ‖X‖ 2 F , X1 = 0
l=1
(823)
where ∑ Φ ij ∈ S N + (796), therefore convex in vecX . We will find this trace
useful as a heuristic to minimize affine dimension of an unknown list arranged
columnar in X (7.2.2), but it tends to facilitate reconstruction of a list
configuration having least energy; id est, it compacts a reconstructed list by
minimizing total norm-square of the vertices.
By substituting G=−V DV 1 (820) into D(G) (810), assuming X1=0
2
(confer5.6.1)
D = δ ( −V DV 2) 1 1 T + 1δ ( −V DV 2) 1 T ( )
− 2 −V DV
1
2
(824)
These relationships will allow combination of distance and Gram
constraints in any optimization problem we may pose:
Constraining all main diagonal entries of a Gram matrix to 1, for
example, is equivalent to the constraint that all points lie on a
hypersphere (5.9.1.0.3) of radius 1 centered at the origin. This
is equivalent to the EDM constraint: D1 = 2N1. [78, p.116] Any
further constraint on that Gram matrix then applies only to interpoint
angle Ψ .
More generally, interpoint angle Ψ can be constrained by fixing all the
individual point lengths δ(G) 1/2 ; then
Ψ = − 1 2 δ2 (G) −1/2 V DV δ 2 (G) −1/2 (825)