v2010.10.26 - Convex Optimization

6.1. DEFINING EDM CONE 4996.1 Defining EDM coneWe invoke a popular matrix criterion to illustrate correspondence between theEDM and PSD cones belonging to the ambient space of symmetric matrices:{−V DV ∈ SND ∈ EDM N+⇔(914)D ∈ S N hwhere V ∈ S N is the geometric centering matrix (B.4). The set of all EDMsof dimension N ×N forms a closed convex cone EDM N because any pair ofEDMs satisfies the definition of a convex cone (175); videlicet, for each andevery ζ 1 , ζ 2 ≥ 0 (A.3.1.0.2)ζ 1 V D 1 V + ζ 2 V D 2 V ≽ 0ζ 1 D 1 + ζ 2 D 2 ∈ S N h⇐ V D 1V ≽ 0, V D 2 V ≽ 0D 1 ∈ S N h , D 2 ∈ S N h(1176)and convex cones are invariant to inverse linear transformation [307, p.22].6.1.0.0.1 Definition. Cone of Euclidean distance matrices.In the subspace of symmetric matrices, the set of all Euclidean distancematrices forms a unique immutable pointed closed convex cone called theEDM cone: for N > 0EDM N { }D ∈ S N h | −V DV ∈ S N += ⋂ {D ∈ S N | 〈zz T , −D〉≥0, δ(D)=0 } (1177)z∈N(1 T )The EDM cone in isomorphic R N(N+1)/2 [sic] is the intersection of an infinitenumber (when N >2) of halfspaces about the origin and a finite numberof hyperplanes through the origin in vectorized variable D = [d ij ] . HenceEDM N is not full-dimensional with respect to S N because it is confined to thesymmetric hollow subspace S N h . The EDM cone relative interior comprisesrel int EDM N = ⋂ {D ∈ S N | 〈zz T , −D〉>0, δ(D)=0 }z∈N(1 T )= { D ∈ EDM N | rank(V DV ) = N −1 } (1178)while its relative boundary comprisesrel∂EDM N = { D ∈ EDM N | 〈zz T , −D〉 = 0 for some z ∈ N(1 T ) }= { D ∈ EDM N | rank(V DV ) < N −1 } (1179)△