v2007.11.26 - Convex Optimization

434 CHAPTER 6. CONE OF DISTANCE MATRICESIn fact, the smallest face that contains auxiliary matrix V of the PSDcone S N + is the intersection with the geometric center subspace (1792) (1793);F ( S N + ∋V ) = cone { V N υυ T V T N= S N c ∩ S N +In isometrically isomorphic R N(N+1)/2| υ ∈ RN−1}(1073)svec F ( S N + ∋V ) = cone T (1074)related to S N c byaff cone T = svec S N c (1075)6.7.2 EDM criteria in 11 T(confer6.5) Laurent specifies an elliptope trajectory condition for EDM conemembership: [172,2.3]D ∈ EDM N ⇔ [1 − e −αd ij] ∈ EDM N ∀α > 0 (925)From the parametrized elliptope E N tD ∈ EDM N ⇔ ∃ t∈ R +E∈ E N tin6.6.2 we propose} D = t11 T − E (1076)Chabrillac & Crouzeix [53,4] prove a different criterion they attributeto Finsler (1937) [97]. We apply it to EDMs: for D ∈ S N h (874)−V T N DV N ≻ 0 ⇔ ∃κ>0 −D + κ11 T ≻ 0⇔D ∈ EDM N with corresponding affine dimension r=N −1(1077)This Finsler criterion has geometric interpretation in terms of thevectorization & projection already discussed in connection with (1067). Withreference to Figure 108, 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.