v2009.01.01 - Convex Optimization

6.8. DUAL EDM CONE 489
6.8.1.7 Schoenberg criterion is discretized membership relation
We show the Schoenberg criterion
−VN TDV }
N ∈ S N−1
+
D ∈ S N h
⇔ D ∈ EDM N (817)
to be a discretized membership relation (2.13.4) between a closed convex
cone K and its dual K ∗ like
〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) ⇔ x ∈ K (329)
where G(K ∗ ) is any set of generators whose conic hull constructs closed
convex dual cone K ∗ :
The Schoenberg criterion is the same as
}
〈zz T , −D〉 ≥ 0 ∀zz T | 11 T zz T = 0
⇔ D ∈ EDM N (1140)
D ∈ S N h
which, by (1141), 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 (1195)
where the zz T constitute a set of generators G for the positive semidefinite
cone’s smallest face F ( S N + ∋V ) (6.7.1) that contains auxiliary matrix V .
From the aggregate in (1153) we get the ordinary membership relation,
assuming only D ∈ S N [173, p.58]
〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ EDM N∗ ⇔ D ∈ EDM N
(1196)
〈D ∗ , D〉 ≥ 0 ∀D ∗ ∈ {δ(u) | u∈ R N } − cone { V N υυ T V T N | υ ∈ RN−1} ⇔ D ∈ EDM N
Discretization (329) yields:
〈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
(1197)