v2006.03.09 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 173The penultimate inequality follows from the generalized inequality andmembership corollary, while the last inequality is a consequence of thatcorollary’s discretization (2.13.4.2.1). 2.56 From (323) and (311) we deduceK ∗ ∩ aff K = cone(X †T ) = {X †T c | c ≽ 0} ⊆ R n (324)is the vertex-description for that section of K ∗ in the affine hull of K becauseR(X †T ) = R(X) by definition of the pseudoinverse. From (253), we knowK ∗ ∩ aff K must be pointed if rel int K is logically assumed nonempty withrespect to aff K .Conversely, suppose full-rank skinny-or-square matrix[ ]X †T =∆ Γ ∗ 1 Γ ∗ 2 · · · Γ ∗ N ∈ R n×N (325)comprises the extreme directions {Γ ∗ i } ⊂ aff K of the dual-cone intersectionwith the affine hull of K . 2.57 From the discrete membership theorem and(257) we get a partial dual to (311); id est, assuming x∈aff cone X{ }x ∈ K ⇔ γ ∗T x ≥ 0 for all γ ∗ ∈ Γ ∗ i , i=1... N ⊂ ∂K ∗ ∩ aff K (326)⇔ X † x ≽ 0 (327)that leads to a partial halfspace-description,K = { x∈aff cone X | X † x ≽ 0 } (328)For γ ∗ =X †T e i , any x =Xa , and for all i we have e T i X † Xa = e T i a ≥ 0only when a ≽ 0. Hence x∈ K .2.56a ≽ 0 ⇔ a T X T X †T c ≥ 0 ∀(c ≽ 0 ⇔ a T X T X †T c ≥ 0 ∀a ≽ 0)∀(c ≽ 0 ⇔ Γ T i X†T c ≥ 0 ∀i) Intuitively, any nonnegative vector a is a conic combination of the standard basis{e i ∈ R N }; a≽0 ⇔ a i e i ≽0 for all i. The last inequality in (323) is a consequence of thefact that x=Xa may be any extreme direction of K , in which case a is a standard basisvector; a = e i ≽ 0. Theoretically, because c≽0 defines a pointed polyhedral cone (in fact,the nonnegative orthant in R N ), we can take (323) one step further by discretizing c :a ≽ 0 ⇔ Γ T i Γ ∗ j ≥ 0 for i,j =1... N ⇔ X † X ≥ 0In words, X † X must be a matrix whose entries are each nonnegative.2.57 When closed convex cone K has empty interior, K ∗ has no extreme directions.