v2007.09.13 - Convex Optimization

426 CHAPTER 6. EDM CONE6.8.1.3 Dual Euclidean distance matrix criterionConditions necessary for membership of a matrix D ∗ ∈ S N to the dualEDM cone EDM N∗ may be derived from (1050): D ∗ ∈ EDM N∗ ⇒D ∗ = δ(y) − V N AVNT for some vector y and positive semidefinite matrixA∈ S N−1+ . This in turn implies δ(D ∗ 1)=δ(y) . Then, for D ∗ ∈ S Nwhere, for any symmetric matrix D ∗D ∗ ∈ EDM N∗ ⇔ δ(D ∗ 1) − D ∗ ≽ 0 (1076)δ(D ∗ 1) − D ∗ ∈ S N c (1077)To show sufficiency of the matrix criterion in (1076), recall Gram-formEDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G (717)where Gram matrix G is positive semidefinite by definition, and recall theself-adjointness property of the main-diagonal linear operator δ (A.1):〈D , D ∗ 〉 = 〈 δ(G)1 T + 1δ(G) T − 2G , D ∗〉 = 〈G , δ(D ∗ 1) − D ∗ 〉 2 (736)Assuming 〈G , δ(D ∗ 1) − D ∗ 〉≥ 0 (1281), then we have known membershiprelation (2.13.2.0.1)D ∗ ∈ EDM N∗ ⇔ 〈D , D ∗ 〉 ≥ 0 ∀D ∈ EDM N (1078)Elegance of this matrix criterion (1076) for membership to the dualEDM cone is the lack of any other assumptions except D ∗ be symmetric.(Recall: Schoenberg criterion (724) for membership to the EDM cone requiresmembership to the symmetric hollow subspace.)Linear Gram-form EDM operator (717) has adjoint, for Y ∈ S NThen we have: [61, p.111]D T (Y ) ∆ = (δ(Y 1) − Y ) 2 (1079)EDM N∗ = {Y ∈ S N | δ(Y 1) − Y ≽ 0} (1080)the dual EDM cone expressed in terms of the adjoint operator. A dual EDMcone determined this way is illustrated in Figure 107.