v2007.09.13 - Convex Optimization

6.8. DUAL EDM CONE 4316.8.1.7 Schoenberg criterion is discretized membership relationWe show the Schoenberg criterion−VN TDV }N ∈ S N−1+D ∈ S N h⇔ D ∈ EDM N (724)to be a discretized membership relation (2.13.4) between a closed convexcone K and its dual K ∗ like〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) ⇔ x ∈ K (317)where G(K ∗ ) is any set of generators whose conic hull constructs closedconvex dual cone K ∗ :The Schoenberg criterion is the same as}〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0⇔ D ∈ EDM N (1037)D ∈ S N hwhich, by (1038), is the same as〈zz T , −D〉 ≥ 0 ∀zz T ∈ { V N υυ T VN T | υ ∈ RN−1}D ∈ S N h}⇔ D ∈ EDM N (1092)where the zz T constitute a set of generators G for the positive semidefinitecone’s smallest face F ( S N + ∋V ) (6.7.1) that contains auxiliary matrix V .From the aggregate in (1050) we get the ordinary membership relation,assuming only D ∈ S N [147, p.58],〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ EDM N∗ ⇔ D ∈ EDM N(1093)〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {δ(u) | u∈ R N } − cone { V N υυ T V T N | υ ∈ RN−1} ⇔ D ∈ EDM NDiscretization yields (317):〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {e i e T i , −e j e T j , −V N υυ T V T N | i, j =1... N , υ ∈ R N−1 } ⇔ D ∈ EDM N(1094)