v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 159
Whenever K is known to be closed and convex, then the converse must
also hold; id est, given any set of generators for K ∗ arranged columnar
in Y , then the consequent vertex-description of the dual cone connotes a
halfspace-description for K : [286,2.8]
K ∗ = {Y a | a ≽ 0} ⇔ K ∗∗ = K = { z | Y T z ≽ 0 } (328)
2.13.4.2 First dual-cone formula
From these two results (327) and (328) we deduce a general principle:
From any [sic] given vertex-description of a convex cone K , a
halfspace-description of the dual cone K ∗ is immediate by matrix
transposition; conversely, from any given halfspace-description, a dual
vertex-description is immediate.
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 also
polyhedral and K ∗∗ = K . [286,2.8]
The generalized inequality and membership corollary is discretized in the
following theorem [20,1] 2.59 that follows directly from (327) and (328):
2.13.4.2.1 Theorem. Discretized membership. (confer2.13.2.0.1)
Given any set of generators (2.8.1.2) denoted by G(K) for closed convex
cone K ⊆ R n and any set of generators denoted G(K ∗ ) for its dual, then
discretization of the generalized inequality and membership corollary (p.152)
is necessary and sufficient for certifying cone membership: for x and y in
vector space R n x ∈ K ⇔ 〈γ ∗ , x〉 ≥ 0 for all γ ∗ ∈ G(K ∗ ) (329)
y ∈ K ∗ ⇔ 〈γ , y〉 ≥ 0 for all γ ∈ G(K) (330)
2.59 Barker & Carlson state the theorem only for the pointed closed convex case.
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