v2007.09.13 - Convex Optimization

364 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXFrom (919) and (386) we get a halfspace-description:K ∗ λ = {y ∈ R N+1 | (V T N [ÂT ˆBT ]) † V T N y ≽ 0} (924)This polyhedral dual spectral cone K ∗ λ is closed, convex, has nonemptyinterior because K λ is pointed, but is not pointed because K λ has emptyinterior.5.11.2.3 Unordered eigenspectraSpectral cones are not unique; eigenspectra ordering can be rendered benignwithin a cone by presorting a vector of eigenvalues into nonincreasingorder. 5.49 Then things simplify: Conditions (910) now specify eigenvaluemembership to the spectral cone([ ]) [ ]0 1T RNλ1 −EDM N += ∩ ∂HR − (925)= {ζ ∈ R N+1 | Bζ ≽ 0, 1 T ζ = 0}where B is defined in (917), and ∂H in (915). From (385) we get avertex-description for a pointed spectral cone having empty interior:([ ])0 1T {λ1 −EDM N = V N ( ˜BV}N ) † b | b ≽ 0{[ ] } (926)I=−1 T b | b ≽ 0where V N ∈ R N+1×N and˜B ∆ =⎡⎢⎣e T 1e T2 .e T N⎤⎥⎦ ∈ RN×N+1 (927)holds only those rows of B corresponding to conically independent rowsin BV N .5.49 Eigenspectra ordering (represented by a cone having monotone description such as(914)) becomes benign in (1134), for example, where projection of a given presorted vectoron the nonnegative orthant in a subspace is equivalent to its projection on the monotonenonnegative cone in that same subspace; equivalence is a consequence of presorting.