v2007.09.13 - Convex Optimization

120 CHAPTER 2. CONVEX GEOMETRYWhen the intersection A ∩ S N + is known a priori to consist only of asingle point, then Barvinok’s proposition provides the greatest upper boundon its rank not exceeding N . The intersection can be a single nonzero pointonly if the number of linearly independent hyperplanes m constituting Asatisfies 2.37 N(N + 1)/2 − 1 ≤ m ≤ N(N + 1)/2 (238)2.10 Conic independence (c.i.)In contrast to extreme direction, the property conically independent directionis more generally applicable, inclusive of all closed convex cones (notonly pointed closed convex cones). Similar to the definition for linearindependence, arbitrary given directions {Γ i ∈ R n , i=1... N} are conicallyindependent if and only if, for all ζ ∈ R N +Γ i ζ i + · · · + Γ j ζ j − Γ l ζ l = 0, i≠ · · · ≠j ≠l = 1... N (239)has only the trivial solution ζ =0; in words, iff no direction from the givenset can be expressed as a conic combination of those remaining. (Figure 37,for example. A Matlab implementation of test (239) is given inF.2.) Itis evident that linear independence (l.i.) of N directions implies their conicindependence;l.i. ⇒ c.i.Arranging any set of generators for a particular convex cone in a matrixcolumnar,X ∆ = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (240)then the relationship l.i. ⇒ c.i. suggests: the number of l.i. generators inthe columns of X cannot exceed the number of c.i. generators. Denoting byk the number of conically independent generators contained in X , we havethe most fundamental rank inequality for convex conesdim aff K = dim aff[0 X ] = rankX ≤ k ≤ N (241)Whereas N directions in n dimensions can no longer be linearly independentonce N exceeds n , conic independence remains possible:2.37 For N >1, N(N+1)/2 −1 independent hyperplanes in R N(N+1)/2 can make a line