v2007.09.13 - Convex Optimization

6.8. DUAL EDM CONE 4276.8.1.3.1 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 (1071) for the dual EDM cone together with the matrixcriterion (1076) for dual EDM cone membership. For any D ∗ ∈ S N it isobvious:δ(D ∗ 1) ∈ S N⊥h (1081)any diagonal matrix belongs to the subspace of diagonal matrices (57). Weknow when D ∗ ∈ EDM N∗ δ(D ∗ 1) − D ∗ ∈ S N c ∩ S N + (1082)this adjoint expression (1079) belongs to that face (1043) 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∗ (1083)can therefore be expressed as the difference of two linearly independentnonorthogonal components (Figure 85, Figure 106).6.8.1.5 Affine dimension complementarityFrom6.8.1.3 we have, for some A∈ S N−1+ (confer (1082))δ(D ∗ 1) − D ∗ = V N AV T N ∈ S N c ∩ S N + (1084)if and only if D ∗ belongs to the dual EDM cone. Call rank(V N AV T N ) dualaffine dimension. Empirically, we find a complementary relationship in affinedimension between the projection of some arbitrary symmetric matrix H onthe polar EDM cone, EDM N◦ = −EDM N∗ , and its projection on the EDMcone; id est, the optimal solution of 6.10minimize ‖D ◦ − H‖ FD ◦ ∈ S Nsubject to D ◦ − δ(D ◦ 1) ≽ 0(1085)6.10 This dual projection can be solved quickly (without semidefinite programming) via