v2010.10.26 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 141then this test of conic independence (279) may be expressed as a set of linearfeasibility problems: for l = 1... Nfind ζ ∈ R Nsubject to Xζ = Γ lζ ≽ 0ζ l = 0(281)If feasible for any particular l , then the set is not conically independent.To find all conically independent directions from a set via (281), generatorΓ l must be removed from the set once it is found (feasible) conicallydependent on remaining generators in X . So, to continue testing remaininggenerators when Γ l is found to be dependent, Γ l must be discarded frommatrix X before proceeding. A generator Γ l that is instead found conicallyindependent of remaining generators in X , on the other hand, is conicallyindependent of any subset of remaining generators. A c.i. set thus found isnot necessarily unique.It is evident that linear independence (l.i.) of N directions implies theirconic independence;l.i. ⇒ c.i.which suggests, number of l.i. generators in the columns of X cannotexceed number of c.i. generators. Denoting by k the number of conicallyindependent generators contained in X , we have the most fundamental rankinequality for convex conesdim aff K = dim aff[0 X ] = rankX ≤ k ≤ N (282)Whereas N directions in n dimensions can no longer be linearly independentonce N exceeds n , conic independence remains possible:2.10.0.0.1 Table: Maximum number of c.i. directionsdimension n supk (pointed) supk (not pointed)0 0 01 1 22 2 43.∞.∞.