v2010.10.26 - Convex Optimization

536 CHAPTER 6. CONE OF DISTANCE MATRICESHere is an algebraic method to prove nonnegativity: Suppose we aregiven A∈ S N⊥c and B = [B ij ]∈ S N + and A −B ∈ S N h . Then we have, forsome vector u , A = u1 T + 1u T = [A ij ] = [u i + u j ] and δ(B)= δ(A)= 2u .Positive semidefiniteness of B requires nonnegativity A −B ≥ 0 because(e i −e j ) T B(e i −e j ) = (B ii −B ij )−(B ji −B jj ) = 2(u i +u j )−2B ij ≥ 0 (1273)6.8.1.3 Dual Euclidean distance matrix criterionConditions necessary for membership of a matrix D ∗ ∈ S N to the dualEDM cone EDM N∗ may be derived from (1248): 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 (1274)δ(D ∗ 1) − D ∗ ∈ S N c (1275)To show sufficiency of the matrix criterion in (1274), recall Gram-formEDM operatorD(G) = δ(G)1 T + 1δ(G) T − 2G (903)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 (921)Assuming 〈G , δ(D ∗ 1) − D ∗ 〉≥ 0 (1485), then we have known membershiprelation (2.13.2.0.1)D ∗ ∈ EDM N∗ ⇔ 〈D , D ∗ 〉 ≥ 0 ∀D ∈ EDM N (1276)