v2007.09.13 - Convex Optimization

152 CHAPTER 2. CONVEX GEOMETRY2.13.4 Discretization of membership relation2.13.4.1 Dual halfspace-descriptionHalfspace-description of the dual cone is equally simple (and extensibleto an infinite number of generators) as vertex-description (252) for thecorresponding closed convex cone: By definition (258), for X ∈ R n×N as in(240), (confer (246))K ∗ = { y ∈ R n | z T y ≥ 0 for all z ∈ K }= { y ∈ R n | z T y ≥ 0 for all z = Xa , a ≽ 0 }= { y ∈ R n | a T X T y ≥ 0, a ≽ 0 }= { y ∈ R n | X T y ≽ 0 } (315)that follows from the generalized inequality and membership corollary (277).The semi-infinity of tests specified by all z ∈ K has been reduced to a setof generators for K constituting the columns of X ; id est, the test has beendiscretized.Whenever K is known to be closed and convex, then the converse mustalso hold; id est, given any set of generators for K ∗ arranged columnarin Y , then the consequent vertex-description of the dual cone connotes ahalfspace-description for K : [245,2.8]K ∗ = {Y a | a ≽ 0} ⇔ K ∗∗ = K = { z | Y T z ≽ 0 } (316)2.13.4.2 First dual-cone formulaFrom these two results (315) and (316) we deduce a general principle:From any given vertex-description of a convex cone K , ahalfspace-description of the dual cone K ∗ is immediate by matrixtransposition; conversely, from any given halfspace-description, a dualvertex-description is immediate.Various other converses are just a little trickier. (2.13.9)We deduce further: For any polyhedral cone K , the dual cone K ∗ is alsopolyhedral and K ∗∗ = K . [245,2.8]The generalized inequality and membership corollary is discretized in thefollowing theorem [18,1] 2.49 that follows directly from (315) and (316):2.49 Barker & Carlson state the theorem only for the pointed closed convex case.