v2009.01.01 - Convex Optimization

6.3.
√
EDM CONE IS NOT CONVEX 451
N = 4. Relative-angle inequality (1067) together with four Euclidean metric
properties are necessary and sufficient tests for realizability of
tetrahedra. (1068) Albeit relative angles θ ikj (859) are nonlinear
functions of the d ij , relative-angle inequality provides a regular
tetrahedron in R 3 [sic] (Figure 112) bounding angles θ ikj at vertex
x k consistently with EDM 4 . 6.2
Yet were we to employ the procedure outlined in5.14.3 for making
generalized triangle inequalities, then we would find all the necessary and
sufficient d ij -transformations for generating bounding polyhedra consistent
with EDMs of any higher dimension (N > 3).
6.3 √ EDM cone is not convex
For some applications, like a molecular conformation problem (Figure 3,
Figure 103) or multidimensional scaling [87] [307], absolute distance √ d ij
is the preferred variable. Taking square root of the entries in all EDMs D
of dimension N , we get another cone but not a convex cone when N > 3
(Figure 114(b)): [78,4.5.2]
√
EDM N ∆ = { ◦√ D | D ∈ EDM N } (1087)
where ◦√ D is defined like (1086). It is a cone simply because any cone
is completely constituted by rays emanating from the origin: (2.7) Any
given ray {ζΓ∈ R N(N−1)/2 | ζ ≥0} remains a ray under entrywise square root:
{ √ ζΓ∈ R N(N−1)/2 | ζ ≥0}. It is already established that
D ∈ EDM N ⇒ ◦√ D ∈ EDM N (999)
But because of how √ EDM N is defined, it is obvious that (confer5.10)
D ∈ EDM N ⇔ ◦√ √
D ∈ EDM N (1088)
Were √ EDM N convex, then given
◦ √ D 1 , ◦√ D 2 ∈ √ EDM N we would
expect their conic combination ◦√ D 1 + ◦√ D 2 to be a member of √ EDM N .
6.2 Still, property-4 triangle inequalities (967) corresponding to each principal 3 ×3
submatrix of −VN TDV N demand that the corresponding √ d ij belong to a polyhedral
cone like that in Figure 114(b).