v2006.03.09 - Convex Optimization

Chapter 5EDM coneFor N > 3, the cone of EDMs is no longer a circular cone andthe geometry becomes complicated...−Hayden, Wells, Liu, & Tarazaga (1991) [110,3]In the subspace of symmetric matrices S N , we know the convex coneof Euclidean distance matrices EDM N does not intersect the positivesemidefinite (PSD) cone S N + except at the origin, their only vertex; therecan be no positive nor negative semidefinite EDM. (646) [145]EDM N ∩ S N + = 0 (722)Even so, the two convex cones can be related. We prove the new equalityEDM N = S N h ∩ ( )S N⊥c − S N + (810)a resemblance to EDM definition (460) whereS N h∆= { A ∈ S N | δ(A) = 0 } (55)is the symmetric hollow subspace (2.2.3) and whereS N⊥c = {u1 T + 1u T | u∈ R N } (1563)is the orthogonal complement of the geometric center subspace (E.7.2.0.2)S N c∆= {Y ∈ S N | Y 1 = 0} (1561)In5.8.1.7 we show: the Schoenberg criterion (479) for discriminatingEuclidean distance matrices is simply a membership relation (2.13.2)between the EDM cone and its ordinary dual.2001 Jon Dattorro. CO&EDG version 03.09.2006. All rights reserved.Citation: Jon Dattorro, Convex Optimization & Euclidean Distance Geometry,Meboo Publishing USA, 2005.305