v2004.06.19 - Convex Optimization

210 CHAPTER 5. EDM CONEA statement parallel to (601) is therefore, for D ∈ EDM N ,(Theorem 4.7.3.0.1)rank(D) = r + 2 ⇔ y /∈ R(V X ) ( ⇔ 1 T D † 1 = 0 )rank(D) = r + 1 ⇔ y ∈ R(V X ) ( ⇔ 1 T D † 1 ≠ 0 ) (604)5.2.2 Faces of EDM coneExpression (596) has utility constructing the set of all EDMs correspondingto affine dimension r :{D ∈ EDM N | rank(V DV )= r }= { δ(V X V T X )1T + 1δ(V X V T X )T − 2V X V T X | V X ∈ R N×r , V T X V X = δ2 (V T X V X ), R(V X)⊆ N(1 T ) }(605)whereas {D ∈ EDM N | rank(V DV )≤ r} is just the closure of this same set;{D ∈ EDM N | rank(V DV )≤ r } = { D ∈ EDM N | rank(V DV )= r } (606)None of these are necessarily convex sets.When cardinality N = 3 and affine dimension r = 2, for example, theinterior rel int EDM 3 is realized via (605). (§5.4)When N = 3 and r = 1, the relative boundary of the EDM conedvec ∂EDM 3 is realized in isomorphic R 3 as in Figure 5.1(d). This figurecould be constructed via (606) by spiraling vector V X tightly about the originin N(1 T ) ; as can be imagined with aid of Figure 5.2. Vectors closeto 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 orbitsthe axis of revolution while remaining on the boundary of the circular conedvec ∂EDM 3 . (Figure 5.3)5.2.2.1 Isomorphic facesIn high cardinality N , any set of EDMs constructed via (605) or (606) withparticular affine dimension r is isomorphic with any set constructed in lowercardinality that admits the same affine dimension. We do not prove thathere.