v2010.10.26 - Convex Optimization

176 CHAPTER 2. CONVEX GEOMETRY2.13.4.2 First dual-cone formulaFrom these two results (363) and (364) we deduce a general principle:From any [sic] given vertex-description (103) of closed convex cone K ,a halfspace-description of the dual cone K ∗ is immediate by matrixtransposition (363); conversely, from any given halfspace-description(287) of K , a dual vertex-description is immediate (364). [307, p.122]Various other converses are just a little trickier. (2.13.9,2.13.11)We deduce further: For any polyhedral cone K , the dual cone K ∗ is alsopolyhedral and K ∗∗ = K . [330, p.56]The generalized inequality and membership corollary is discretized in thefollowing theorem inspired by (363) and (364):2.13.4.2.1 Theorem. Discretized membership. (confer2.13.2.0.1) 2.66Given any set of generators, (2.8.1.2) denoted by G(K) for closed convexcone K ⊆ R n , and any set of generators denoted G(K ∗ ) for its dual such thatK = cone G(K) , K ∗ = cone G(K ∗ ) (365)then discretization of the generalized inequality and membership corollary(p.168) is necessary and sufficient for certifying cone membership: for x andy in vector space R nx ∈ K ⇔ 〈γ ∗ , x〉 ≥ 0 for all γ ∗ ∈ G(K ∗ ) (366)y ∈ K ∗ ⇔ 〈γ , y〉 ≥ 0 for all γ ∈ G(K) (367)⋄Proof. K ∗ = {G(K ∗ )a | a ≽ 0}. y ∈ K ∗ ⇔ y=G(K ∗ )a for some a ≽ 0.x∈ K ⇔ 〈y , x〉≥ 0 ∀y ∈ K ∗ ⇔ 〈G(K ∗ )a , x〉≥ 0 ∀a ≽ 0 (320). a ∑ i α ie iwhere e i is the i th member of a standard basis of possibly infinitecardinality. 〈G(K ∗ )a , x〉≥ 0 ∀a ≽ 0 ⇔ ∑ i α i〈G(K ∗ )e i , x〉≥ 0 ∀α i ≥ 0 ⇔〈G(K ∗ )e i , x〉≥ 0 ∀i. Conjugate relation (367) is similarly derived. 2.66 Stated in [21,1] without proof for pointed closed convex case.