v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 191We consider nonempty pointed polyhedral cone K possibly notfull-dimensional; id est, we consider a basis spanning a subspace. Then weneed only observe that section of dual cone K ∗ in the affine hull of K because,by expansion of x , membership x∈aff K is implicit and because any breachof the ordinary dual cone into ambient space becomes irrelevant (2.13.9.3).Biorthogonal expansionx = XX † x ∈ aff K = aff cone(X) (403)is expressed in the extreme directions {Γ i } of K arranged columnar inunder assumption of biorthogonalityX = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (280)X † X = I (404)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 2.72 as the extremedirections {Γ i } ofK = cone(X) = {Xa | a ≽ 0} ⊆ R n (103)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 2.73 -or-square full-rank). In other words, fat full-rankmatrix X is prohibited by uniqueness because of existence of an infinity ofright inverses;polyhedral cones whose extreme directions number in excess of theambient space dimension are precluded in biorthogonal expansion.2.72 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.2.73 “Skinny” meaning thin; more rows than columns.