v2010.10.26 - Convex Optimization

542 CHAPTER 6. CONE OF DISTANCE MATRICESBecause 〈 {δ(u) | u∈ R N }, D 〉 ≥ 0 ⇔ D ∈ S N h , we can restrict observationto the symmetric hollow subspace without loss of generality. Then for D ∈ S N h〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ { −V N υυ T V T N | υ ∈ R N−1} ⇔ D ∈ EDM N (1293)this discretized membership relation becomes (1290); identical to theSchoenberg criterion.Hitherto a correspondence between the EDM cone and a face of a PSDcone, the Schoenberg criterion is now accurately interpreted as a discretizedmembership relation between the EDM cone and its ordinary dual.6.8.2 Ambient S N hWhen instead we consider the ambient space of symmetric hollow matrices(1249), then still we find the EDM cone is not selfdual for N >2. Thesimplest way to prove this is as follows:Given a set of generators G = {Γ} (1209) for the pointed closed convexEDM cone, the discretized membership theorem in2.13.4.2.1 asserts thatmembers of the dual EDM cone in the ambient space of symmetric hollowmatrices can be discerned via discretized membership relation:EDM N∗ ∩ S N h {D ∗ ∈ S N h | 〈Γ , D ∗ 〉 ≥ 0 ∀ Γ ∈ G(EDM N )}By comparison= {D ∗ ∈ S N h | 〈δ(zz T )1 T + 1δ(zz T ) T − 2zz T , D ∗ 〉 ≥ 0 ∀z∈ N(1 T )}= {D ∗ ∈ S N h | 〈1δ(zz T ) T − zz T , D ∗ 〉 ≥ 0 ∀z∈ N(1 T )}(1294)EDM N = {D ∈ S N h | 〈−zz T , D〉 ≥ 0 ∀z∈ N(1 T )} (1295)the term δ(zz T ) T D ∗ 1 foils any hope of selfdualness in ambient S N h . To find the dual EDM cone in ambient S N h per2.13.9.4 we prune theaggregate in (1248) describing the ordinary dual EDM cone, removing anymember having nonzero main diagonal:EDM N∗ ∩ S N h = cone { δ 2 (V N υυ T V T N ) − V N υυT V T N | υ ∈ RN−1}= {δ 2 (V N ΨV T N ) − V N ΨV T N| Ψ∈ SN−1 + }(1296)