v2010.10.26 - Convex Optimization

140 CHAPTER 2. CONVEX GEOMETRYand matrix dimension N ≥ ρ + 2 ≥ 3. If the intersection is nonempty andbounded, then there exists a matrix X ∈ A ∩ S N + such thatrankX ≤ ρ (276)This represents a tightening of the upper bound; a reduction by exactly 1of the bound provided by (272) given the same specified number m (275) ofequalities; id est,rankX ≤√ 8m + 1 − 12− 1 (277)⋄When intersection A ∩ S N + is known a priori to consist only of a singlepoint, then Barvinok’s proposition provides the greatest upper bound onits rank not exceeding N . The intersection can be a single nonzero pointonly if the number of linearly independent hyperplanes m constituting Asatisfies 2.51 N(N + 1)/2 − 1 ≤ m ≤ N(N + 1)/2 (278)2.10 Conic independence (c.i.)In contrast to extreme direction, the property conically independentdirection is more generally applicable; inclusive of all closed convexcones (not only pointed closed convex cones). Arbitrary given directions{Γ i ∈ R n , i=1... N} comprise a conically independent set if and only if(confer2.1.2,2.4.2.3)Γ i ζ i + · · · + Γ j ζ j − Γ l = 0, i≠ · · · ≠j ≠l = 1... N (279)has no solution ζ ∈ R N + ; in words, iff no direction from the given set canbe expressed as a conic combination of those remaining; e.g., Figure 49[test (279) Matlab implementation, Wıκımization]. Arranging any setof generators for a particular closed convex cone in a matrix columnar,X [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (280)2.51 For N >1, N(N+1)/2 −1 independent hyperplanes in R N(N+1)/2 can make a linetangent to svec ∂ S N + at a point because all one-dimensional faces of S N + are exposed.Because a pointed convex cone has only one vertex, the origin, there can be no intersectionof svec ∂ S N + with any higher-dimensional affine subset A that will make a nonzero point.