v2007.09.17 - Convex Optimization

164 CHAPTER 2. CONVEX GEOMETRYunder assumption of biorthogonalityX † X = I (345)where † denotes matrix pseudoinverse (E). We therefore seek, in thissection, a vertex-description for K ∗ ∩ aff K in terms of linearly independentdual generators {Γ ∗ i }⊂aff K in the same finite quantity 2.53 as the extremedirections {Γ i } ofK = cone(X) = {Xa | a ≽ 0} ⊆ R n (252)We assume the quantity of extreme directions N does not exceed thedimension n of ambient vector space because, otherwise, the expansion couldnot be unique; id est, assume N linearly independent extreme directionshence N ≤ n (X skinny 2.54 -or-square full-rank). In other words, fat full-rankmatrix X is prohibited by uniqueness because of the existence of an infinityof right-inverses;polyhedral cones whose extreme directions number in excess of theambient space dimension are precluded in biorthogonal expansion.2.13.8.1 x ∈ KSuppose x belongs to K ⊆ R n . Then x =Xa for some a≽0. Vector a isunique only when {Γ i } is a linearly independent set. 2.55 Vector a∈ R N cantake the form a =Bx if R(B)= R N . Then we require Xa =XBx = x andBx=BXa = a . The pseudoinverse B =X † ∈ R N×n (E) is suitable when Xis skinny-or-square and full-rank. In that case rankX =N , and for all c ≽ 0and i=1... Na ≽ 0 ⇔ X † Xa ≽ 0 ⇔ a T X T X †T c ≥ 0 ⇔ Γ T i X †T c ≥ 0 (346)The penultimate inequality follows from the generalized inequality andmembership corollary, while the last inequality is a consequence of that2.53 When K is contained in a proper subspace of R n , the ordinary dual cone K ∗ will havemore generators in any minimal set than K has extreme directions.2.54 “Skinny” meaning thin; more rows than columns.2.55 Conic independence alone (2.10) is insufficient to guarantee uniqueness.