v2007.09.13 - Convex Optimization

116 CHAPTER 2. CONVEX GEOMETRY2.9.2.6.3 Theorem. Convex subsets of positive semidefinite cone.The subsets of the positive semidefinite cone S M + , for 0≤ρ≤MS M +(ρ) ∆ = {X ∈ S M + | rankX ≥ ρ} (219)are pointed convex cones, but not closed unless ρ = 0 ; id est, S M +(0)= S M + .⋄Proof. Given ρ , a subset S M +(ρ) is convex if and only ifconvex combination of any two members has rank at least ρ . That isconfirmed applying identity (215) from Lemma 2.9.2.6.1 to (218); id est, forA,B ∈ S M +(ρ) on the closed interval µ∈[0, 1]rank(µA + (1 −µ)B) ≥ min{rankA, rankB} (220)It can similarly be shown, almost identically to proof of the lemma, any coniccombination of A,B in subset S M +(ρ) remains a member; id est, ∀ζ, ξ ≥0rank(ζA + ξB) ≥ min{rank(ζA), rank(ξB)} (221)Therefore, S M +(ρ) is a convex cone.Another proof of convexity can be made by projection arguments:2.9.2.7 Projection on S M +(ρ)Because these cones S M +(ρ) indexed by ρ (219) are convex, projection onthem is straightforward. Given a symmetric matrix H having diagonalizationH = ∆ QΛQ T ∈ S M (A.5.2) with eigenvalues Λ arranged in nonincreasingorder, then its Euclidean projection (minimum-distance projection) on S M +(ρ)P S M+ (ρ)H = QΥ ⋆ Q T (222)corresponds to a map of its eigenvalues:Υ ⋆ ii ={ max {ǫ , Λii } , i=1... ρmax {0, Λ ii } , i=ρ+1... M(223)where ǫ is positive but arbitrarily close to 0.