v2007.09.13 - Convex Optimization

166 CHAPTER 2. CONVEX GEOMETRYWhen X is full-rank, then the unique biorthogonal expansion of x ∈ Kbecomes (344)N∑x = XX † x = Γ i Γ ∗Ti x (352)whose coordinates Γ ∗Ti x must be nonnegative because K is assumed pointedclosed and convex. Whenever X is full-rank, so is its pseudoinverse X † .(E) In the present case, the columns of X †T are linearly independent andgenerators of the dual cone K ∗ ∩ aff K ; hence, the columns constitute itsextreme directions. (2.10) That section of the dual cone is itself a polyhedralcone (by (246) or the cone intersection theorem,2.7.2.1.1) having the samenumber of extreme directions as K .2.13.8.2 x ∈ aff KThe extreme directions of K and K ∗ ∩ aff K have a distinct relationship;because X † X = I , then for i,j = 1... N , Γ T i Γ ∗ i = 1, while for i ≠ j ,Γ T i Γ ∗ j = 0. Yet neither set of extreme directions, {Γ i } nor {Γ ∗ i } , isnecessarily orthogonal. This is, precisely, a biorthogonality condition,[273,2.2.4] [149] implying each set of extreme directions is linearlyindependent. (B.1.1.1)The biorthogonal expansion therefore applies more broadly; meaning,for any x ∈ aff K , vector x can be uniquely expressed x =Xb whereb∈R N because aff K contains the origin. Thus, for any such x∈ R(X)(conferE.1.1), biorthogonal expansion (352) becomes x =XX † Xb =Xb .2.13.9 Formulae, algorithm finding dual cone2.13.9.1 Pointed K , dual, X skinny-or-square full-rankWe wish to derive expressions for a convex cone and its ordinary dualunder the general assumptions: pointed polyhedral K denoted by its linearlyindependent extreme directions arranged columnar in matrix X such thatThe vertex-description is given:i=1rank(X ∈ R n×N ) = N ∆ = dim aff K ≤ n (353)K = {Xa | a ≽ 0} ⊆ R n (354)