v2010.10.26 - Convex Optimization

6.5. CORRESPONDENCE TO PSD CONE S N−1+ 511and where {δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i , i=1... N} are extremedirections of some pointed polyhedral cone K ⊂ S N h and extreme directionsof EDM N . Invertibility of (1210)−V DV 1 = −V ∑ N (λ2 i δ(qi qi T )1 T + 1δ(q i qi T ) T − 2q i qiTi=1∑= N λ i q i qiTi=1)V12(1211)implies linear independence of those extreme directions.biorthogonal expansion is expresseddvec D = Y Y † dvecD = Y λ ( )−V DV 1 2Then the(1212)whereY [ dvec ( δ(q i q T i )1 T + 1δ(q i q T i ) T − 2q i q T i), i = 1... N]∈ R N(N−1)/2×N (1213)When D belongs to the EDM cone in the subspace of symmetric hollowmatrices, unique coordinates Y † dvecD for this biorthogonal expansionmust be the nonnegative eigenvalues λ of −V DV 1 . This means D2simultaneously belongs to the EDM cone and to the pointed polyhedral conedvec K = cone(Y ).6.4.3.3 open questionResult (1207) is analogous to that for the positive semidefinite cone (222),although the question remains open whether all faces of EDM N (whosedimension is less than dimension of the cone) are exposed like they are forthe positive semidefinite cone. 6.7 (2.9.2.3) [347]6.5 Correspondence to PSD cone S N−1+Hayden, Wells, Liu, & Tarazaga [185,2] assert one-to-one correspondenceof EDMs with positive semidefinite matrices in the symmetric subspace.Because rank(V DV )≤N−1 (5.7.1.1), that PSD cone corresponding to6.7 Elementary example of face not exposed is given by the closed convex set in Figure 32and in Figure 42.