v2007.09.13 - Convex Optimization

362 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXPositive semidefiniteness of that Schur complement insures nonnegativity(D ∈ R N×N+ ,5.8.1), whereas complementary inertia (1314) insures that lonenegative eigenvalue of the Cayley-Menger form.Now we apply results from chapter 2 with regard to polyhedral cones andtheir duals.5.11.2.2 Ordered eigenspectraConditions (910) specify eigenvalue [ membership ] to the smallest pointed0 1Tpolyhedral spectral cone for1 −EDM N :K ∆ λ = {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N ≥ 0 ≥ ζ N+1 , 1 T ζ = 0}[ ]RN+= K M ∩ ∩ ∂HR −([ ])0 1T= λ1 −EDM N(914)where∂H ∆ = {ζ ∈ R N+1 | 1 T ζ = 0} (915)is a hyperplane through the origin, and K M is the monotone cone(2.13.9.4.2, implying ordered eigenspectra) which has nonempty interior butis not pointed;K M = {ζ ∈ R N+1 | ζ 1 ≥ ζ 2 ≥ · · · ≥ ζ N+1 } (377)K ∗ M = { [e 1 − e 2 e 2 −e 3 · · · e N −e N+1 ]a | a ≽ 0 } ⊂ R N+1 (378)So because of the hyperplane,indicating K λ has empty interior. Defining⎡⎡e T 1 − e T ⎤2A =∆ ⎢e T 2 − e T 3 ⎥⎣ . ⎦ ∈ RN×N+1 , B ∆ =⎢e T N − ⎣eT N+1dim aff K λ = dim ∂H = N (916)e T 1e T2 .e T N−e T N+1⎤⎥⎦ ∈ RN+1×N+1 (917)