v2009.01.01 - Convex Optimization

462 CHAPTER 6. CONE OF DISTANCE MATRICES
6.5.2 Boundary constituents of EDM cone
Expression (1093) has utility in forming the set of all EDMs corresponding
to affine dimension r :
{
D ∈ EDM N | rank(V DV )= r }
= { D(V X ) | V X ∈ R N×r , rankV X =r , V T X V X = δ2 (V T X V X ), R(V X)⊆ N(1 T ) }
whereas {D ∈ EDM N | rank(V DV )≤ r} is the closure of this same set;
(1110)
{
D ∈ EDM N | rank(V DV )≤ r } = { D ∈ EDM N | rank(V DV )= r } (1111)
For example,
rel ∂EDM N = { D ∈ EDM N | rank(V DV )< N −1 }
= N−2
⋃ {
D ∈ EDM N | rank(V DV )= r } (1112)
r=0
None of these are necessarily convex sets, although
EDM N = N−1 ⋃ {
D ∈ EDM N | rank(V DV )= r }
r=0
= { D ∈ EDM N | rank(V DV )= N −1 }
rel int EDM N = { D ∈ EDM N | rank(V DV )= N −1 } (1113)
are pointed convex cones.
When cardinality N = 3 and affine dimension r = 2, for example, the
relative interior rel int EDM 3 is realized via (1110). (6.6)
When N = 3 and r = 1, the relative boundary of the EDM cone
dvec rel∂EDM 3 is realized in isomorphic R 3 as in Figure 114(d). This figure
could be constructed via (1111) by spiraling vector V X tightly about the
origin in N(1 T ) ; as can be imagined with aid of Figure 119. Vectors close
to the origin in N(1 T ) are correspondingly close to the origin in EDM N .
As vector V X orbits the origin in N(1 T ) , the corresponding EDM orbits
the axis of revolution while remaining on the boundary of the circular cone
dvec rel∂EDM 3 . (Figure 120)