v2010.10.26 - Convex Optimization

192 CHAPTER 2. CONVEX GEOMETRY2.13.8.1 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. 2.74 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 (405)The 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). 2.75 From (405) and (393) we deduceK ∗ ∩ aff K = cone(X †T ) = {X †T c | c ≽ 0} ⊆ R n (406)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 (310), 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 (N ≤ n)[ ]X †T Γ ∗ 1 Γ ∗ 2 · · · Γ ∗ N ∈ R n×N (407)comprises the extreme directions {Γ ∗ i } ⊂ aff K of the dual-cone intersectionwith the affine hull of K . 2.76 From the discretized membership theorem and2.74 Conic independence alone (2.10) is insufficient to guarantee uniqueness.2.75a ≽ 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 (405) 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 (405) 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.2.76 When closed convex cone K is not full-dimensional, K ∗ has no extreme directions.