v2010.10.26 - Convex Optimization

6.4. EDM DEFINITION IN 11 T 507Then a statement parallel to (1198) is, for D ∈ EDM N (Theorem 5.7.3.0.1)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 ) (1201)6.4.2 Boundary constituents of EDM coneExpression (1185) has utility in forming the set of all EDMs correspondingto 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 ) }(1202)whereas {D ∈ EDM N | rank(V DV )≤ r} is the closure of this same set;{D ∈ EDM N | rank(V DV )≤ r } = { D ∈ EDM N | rank(V DV )= r } (1203)For example,rel ∂EDM N = { D ∈ EDM N | rank(V DV )< N −1 }= N−2⋃ {D ∈ EDM N | rank(V DV )= r } (1204)r=0None of these are necessarily convex sets, althoughEDM 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 } (1205)are pointed convex cones.When cardinality N = 3 and affine dimension r = 2, for example, therelative interior rel int EDM 3 is realized via (1202). (6.5)When N = 3 and r = 1, the relative boundary of the EDM conedvec rel∂EDM 3 is realized in isomorphic R 3 as in Figure 138d. This figurecould be constructed via (1203) by spiraling vector V X tightly about theorigin in N(1 T ) ; as can be imagined with aid of Figure 139. 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 rel∂EDM 3 . (Figure 140)