v2006.03.09 - Convex Optimization

5.7. VECTORIZATION & PROJECTION INTERPRETATION 333svec ∂ S 2 +[ ]d11 d 12d 12 d 22d 22svec EDM 20−Td 11√2d12T ∆ ={[ 1svec(zz T ) | z ∈ N(11 T )= κ−1] }, κ∈ R ⊂ svec ∂ S 2 +Figure 82: Truncated boundary of positive semidefinite cone S 2 + inisometrically isomorphic R 3 (via svec (46)) is, in this dimension, constitutedsolely by its extreme directions. Truncated cone of Euclidean distancematrices EDM 2 in isometrically isomorphic subspace R . Relativeboundary of EDM cone is constituted solely by matrix 0. HalflineT = {κ 2 [ 1 − √ 2 1 ] T | κ∈ R} on PSD cone boundary depicts that loneextreme ray (780) on which orthogonal projection of −D must be positivesemidefinite if D is to belong to EDM 2 . aff cone T = svec S 2 c . (785) DualEDM cone is halfspace in R 3 whose bounding hyperplane has inward-normalsvec EDM 2 .