v2009.01.01 - Convex Optimization

412 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
length-N list because every convex polyhedron having N or fewer vertices
can be generated that way (2.12.2). Equivalent to (985) are
{p T p | p ∈ P − α} = {p T p = y T Θ T pΘ p y | y ∈ S − β} (996)
Because p ∈ P − α may be found by factoring (996), the list Θ p is found
by factoring (995). A unique EDM can be made from that list using
inner-product form definition D(Θ)| Θ=Θp (863). That EDM will be identical
to D if δ(D)=0, by injectivity of D (914).
5.9.2 Necessity and sufficiency
From (956) we learned that matrix inequality −VN TDV N ≽ 0 is a necessary
test for D to be an EDM. In5.9.1, the connection between convex polyhedra
and EDMs was pronounced by the EDM assertion; the matrix inequality
together with D ∈ S N h became a sufficient test when the EDM assertion
demanded that every bounded convex polyhedron have a corresponding
EDM. For all N >1 (5.8.3), the matrix criteria for the existence of an EDM
in (817), (980), and (793) are therefore necessary and sufficient and subsume
all the Euclidean metric properties and further requirements.
5.9.3 Example revisited
Now we apply the necessary and sufficient EDM criteria (817) to an earlier
problem.
5.9.3.0.1 Example. Small completion problem, III. (confer5.8.3.1.1)
Continuing Example 5.3.0.0.2 pertaining to Figure 93 where N = 4,
distance-square d 14 is ascertainable from the matrix inequality −VN TDV N ≽0.
Because all distances in (790) are known except √ d 14 , we may simply
calculate the smallest eigenvalue of −VN TDV N over a range of d 14 as in
Figure 108. We observe a unique value of d 14 satisfying (817) where the
abscissa is tangent to the hypograph of the smallest eigenvalue. Since
the smallest eigenvalue of a symmetric matrix is known to be a concave
function (5.8.4), we calculate its second partial derivative with respect to
d 14 evaluated at 2 and find −1/3. We conclude there are no other satisfying
values of d 14 . Further, that value of d 14 does not meet an upper or lower
bound of a triangle inequality like (968), so neither does it cause the collapse