v2010.10.26 - Convex Optimization

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 137corresponds to a map of its eigenvalues:Υ ⋆ ii ={ max {ǫ , Λii } , i=1... ρmax {0, Λ ii } , i=ρ+1... M(264)where ǫ is positive but arbitrarily close to 0.2.9.2.10.1 Exercise. Projection on open convex cones.Prove (264) using Theorem E.9.2.0.1.Because each H ∈ S M has unique projection on S M +(ρ) (despite possibilityof repeated eigenvalues in Λ), we may conclude it is a convex set by theBunt-Motzkin theorem (E.9.0.0.1).Compare (264) to the well-known result regarding Euclidean projectionon a rank ρ subset of the positive semidefinite cone (2.9.2.1)S M + \ S M +(ρ + 1) = {X ∈ S M + | rankX ≤ ρ} (216)P S M+ \S M + (ρ+1) H = QΥ ⋆ Q T (265)As proved in7.1.4, this projection of H corresponds to the eigenvalue mapΥ ⋆ ii ={ max {0 , Λii } , i=1... ρ0 , i=ρ+1... M(1341)Together these two results (264) and (1341) mean: A higher-rank solutionto projection on the positive semidefinite cone lies arbitrarily close to anygiven lower-rank projection, but not vice versa. Were the number ofnonnegative eigenvalues in Λ known a priori not to exceed ρ , then thesetwo different projections would produce identical results in the limit ǫ→0.2.9.2.11 Uniting constituentsInterior of the PSD cone int S M + is convex by Theorem 2.9.2.9.3, for example,because all positive semidefinite matrices having rank M constitute the coneinterior.All positive semidefinite matrices of rank less than M constitute the coneboundary; an amalgam of positive semidefinite matrices of different rank.Thus each nonconvex subset of positive semidefinite matrices, for 0