v2010.10.26 - Convex Optimization

6.8. DUAL EDM CONE 537Elegance of this matrix criterion (1274) for membership to the dual EDMcone is lack of any other assumptions except D ∗ be symmetric: 6.12 LinearGram-form EDM operator D(Y ) (903) has adjoint, for Y ∈ S NThen from (1276) and (904) we have: [89, p.111]D T (Y ) (δ(Y 1) − Y ) 2 (1277)EDM N∗ = {D ∗ ∈ S N | 〈D , D ∗ 〉 ≥ 0 ∀D ∈ EDM N }= {D ∗ ∈ S N | 〈D(G), D ∗ 〉 ≥ 0 ∀G ∈ S N +}= {D ∗ ∈ S N | 〈 G, D T (D ∗ ) 〉 ≥ 0 ∀G ∈ S N +}= {D ∗ ∈ S N | δ(D ∗ 1) − D ∗ ≽ 0}(1278)the dual EDM cone expressed in terms of the adjoint operator. A dual EDMcone determined this way is illustrated in Figure 151.6.8.1.3.1 Exercise. Dual EDM spectral cone.Find a spectral cone as in5.11.2 corresponding to EDM N∗ .6.8.1.4 Nonorthogonal components of dual EDMNow we tie construct (1269) for the dual EDM cone together with the matrixcriterion (1274) for dual EDM cone membership. For any D ∗ ∈ S N it isobvious:δ(D ∗ 1) ∈ S N⊥h (1279)any diagonal matrix belongs to the subspace of diagonal matrices (67). Weknow when D ∗ ∈ EDM N∗ δ(D ∗ 1) − D ∗ ∈ S N c ∩ S N + (1280)this adjoint expression (1277) belongs to that face (1238) of the positivesemidefinite cone S N + in the geometric center subspace. Any nonzerodual EDMD ∗ = δ(D ∗ 1) − (δ(D ∗ 1) − D ∗ ) ∈ S N⊥h ⊖ S N c ∩ S N + = EDM N∗ (1281)can therefore be expressed as the difference of two linearly independent (whenvectorized) nonorthogonal components (Figure 129, Figure 150).6.12 Recall: Schoenberg criterion (910) for membership to the EDM cone requiresmembership to the symmetric hollow subspace.