v2010.10.26 - Convex Optimization

5.11. EDM INDEFINITENESS 473So because of the hyperplane,indicating K λ is not full-dimensional. Defining⎡⎡e T 1 − e T ⎤2A ⎢e T 2 − e T 3 ⎥⎣ . ⎦ ∈ RN×N+1 , B ⎢e T N − ⎣eT N+1we have the halfspace-description:dim aff K λ = dim∂H = N (1113)e T 1e T2 .e T N−e T N+1⎤⎥⎦ ∈ RN+1×N+1 (1114)K λ = {ζ ∈ R N+1 | Aζ ≽ 0, Bζ ≽ 0, 1 T ζ = 0} (1115)From this and (444) we get a vertex-description for a pointed spectral conethat is not full-dimensional:{ ([ ] ) †}ÂK λ = V N V N b | b ≽ 0(1116)ˆBwhere V N ∈ R N+1×N , and where [sic]ˆB = e T N ∈ R 1×N+1 (1117)and⎡e T 1 − e T ⎤2Â = ⎢e T 2 − e T 3 ⎥⎣ . ⎦ ∈ RN−1×N+1 (1118)e T N−1 − eT Nhold those[ rows]of A and B corresponding to conically independent rowsA(2.10) in VB N .Conditions (1107) can be equivalently restated in terms of a spectral conefor Euclidean distance matrices:⎧ ([ ]) [ ]⎪⎨ 0 1TRN+λ∈ KD ∈ EDM N ⇔ 1 −DM ∩ ∩ ∂HR − (1119)⎪⎩D ∈ S N h