v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 381
because (A.3.1.0.5)
Ω ≽ 0 ⇒ Θ T Θ ≽ 0 (871)
Decomposition (868) and the relative-angle matrix inequality Ω ≽ 0 lead to
a different expression of an inner-product form EDM definition (863)
D(Ω,d) ∆ =
=
[ ] 0
1
d
T + 1 [ 0 d ] √ ([ ])[ ] √ ([ ])
0 0 0 T T 0
− 2 δ
δ
d 0 Ω d
[ ]
0 d
T
d d1 T + 1d T − 2 √ δ(d) Ω √ ∈ EDM N
δ(d)
(872)
and another expression of the EDM cone:
EDM N =
{
D(Ω,d) | Ω ≽ 0, √ }
δ(d) ≽ 0
(873)
In the particular circumstance x 1 = 0, we can relate interpoint angle
matrix Ψ from the Gram decomposition in (807) to relative-angle matrix
Ω in (868). Thus,
[ ] 1 0
T
Ψ ≡ , x
0 Ω 1 = 0 (874)
5.4.3.2 Inner-product form −V T N D(Θ)V N convexity
On
[√
page 379
]
we saw that each EDM entry d ij is a convex quadratic function
dik
of √dkj and a quasiconvex function of θ ikj . Here the situation for
inner-product form EDM operator D(Θ) (863) is identical to that in5.4.1
for list-form D(X) ; −D(Θ) is not a quasiconvex function of Θ by the same
reasoning, and from (867)
−V T N D(Θ)V N = Θ T Θ (875)
is a convex quadratic function of Θ on domain R n×N−1 achieving its minimum
at Θ = 0.