v2010.10.26 - Convex Optimization

6.6. VECTORIZATION & PROJECTION INTERPRETATION 521In fact the smallest face, that contains auxiliary matrix V , of the PSDcone S N + is the intersection with the geometric center subspace (1998) (1999);F ( S N + ∋V ) = cone { V N υυ T VNT | υ ∈ RN−1}= S N c ∩ S N +≡ {X ≽ 0 | 〈X , 11 T 〉 = 0} (1593)(1238)In isometrically isomorphic R N(N+1)/2svec F ( S N + ∋V ) = cone T (1239)related to S N c byaff cone T = svec S N c (1240)6.6.2 EDM criteria in 11 T(confer6.4, (917)) Laurent specifies an elliptope trajectory condition forEDM cone membership: [235,2.3]D ∈ EDM N ⇔ [1 − e −αd ij] ∈ EDM N ∀α > 0 (1092a)From the parametrized elliptope E N tD ∈ EDM N ⇔ ∃ t∈ R +E∈ E N tin6.5.2 and5.10.1 we propose} D = t11 T − E (1241)Chabrillac & Crouzeix [74,4] prove a different criterion they attributeto Finsler (1937) [144]. We apply it to EDMs: for D ∈ S N h (1040)−V T N DV N ≻ 0 ⇔ ∃κ>0 −D + κ11 T ≻ 0⇔D ∈ EDM N with corresponding affine dimension r=N −1(1242)This Finsler criterion has geometric interpretation in terms of thevectorization & projection already discussed in connection with (1232). Withreference to Figure 142, the offset 11 T is simply a direction orthogonal toT in isomorphic R 3 . Intuitively, translation of −D in direction 11 T is likeorthogonal projection on T in so far as similar information can be obtained.