v2010.10.26 - Convex Optimization

136 CHAPTER 2. CONVEX GEOMETRYRank is a quasiconcave function on S M + because the right-hand inequality in(1466) has the concave form (643); videlicet, Lemma 2.9.2.9.1. From this example we see, unlike convex functions, quasiconvex functionsare not necessarily continuous. (3.8) We also glean:2.9.2.9.3 Theorem. Convex subsets of positive semidefinite cone.Subsets of the positive semidefinite cone S M + , for 0≤ρ≤MS M +(ρ) {X ∈ S M + | rankX ≥ ρ} (260)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 if convexcombination of any two members has rank at least ρ . That is confirmedby applying identity (257) from Lemma 2.9.2.9.1 to (1466); id est, forA,B ∈ S M +(ρ) on closed interval [0, 1] of µrank(µA + (1 −µ)B) ≥ min{rankA, rankB} (261)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)} (262)Therefore, S M +(ρ) is a convex cone.Another proof of convexity can be made by projection arguments:2.9.2.10 Projection on S M +(ρ)Because these cones S M +(ρ) indexed by ρ (260) are convex, projection onthem is straightforward. Given a symmetric matrix H having diagonalizationH QΛQ T ∈ S M (A.5.1) 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 (263)