v2006.03.09 - Convex Optimization

130 CHAPTER 2. CONVEX GEOMETRY000(a) (b) (c)Figure 35: Vectors in R 2 : (a) affinely and conically independent,(b) affinely independent but not conically independent, (c) conicallyindependent but not affinely independent. None of the examples exhibitslinear independence. (In general, a.i. c.i.)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 35,for example. A Matlab implementation of test (227) is given inG.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 (228)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 (229)Whereas N directions in n dimensions can no longer be linearly independentonce N exceeds n , conic independence remains possible: