v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 175is the feasible set where Z ∈ R n×n−rank C holds basis N(C) columnar, andx p is any particular solution to Cx = d . Since x ⋆ ∈ C , we arbitrarily choosex p = x ⋆ which yields an equivalent optimality condition∇f(x ⋆ ) T Zξ ≥ 0 ∀ξ∈ R n−rank C (360)when substituted into (352). But this is simply half of a membership relationwhere the cone dual to R n−rank C is the origin in R n−rank C . We must thereforehaveZ T ∇f(x ⋆ ) = 0 ⇔ ∇f(x ⋆ ) T Zξ ≥ 0 ∀ξ∈ R n−rank C (361)meaning, ∇f(x ⋆ ) must be orthogonal to N(C). These conditionsare necessary and sufficient for optimality.Z T ∇f(x ⋆ ) = 0, Cx ⋆ = d (362)2.13.4 Discretization of membership relation2.13.4.1 Dual halfspace-descriptionHalfspace-description of dual cone K ∗ is equally simple as vertex-descriptionK = cone(X) = {Xa | a ≽ 0} ⊆ R n (103)for corresponding closed convex cone K : By definition (297), for X ∈ R n×Nas in (280), (confer (287))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 } (363)that follows from the generalized inequality and membership corollary (321).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 cone K is known to be closed and convex, the conjugatestatement must also hold; id est, given any set of generators for dual coneK ∗ arranged columnar in Y , then the consequent vertex-description of dualcone connotes a halfspace-description for K : [330,2.8]K ∗ = {Y a | a ≽ 0} ⇔ K ∗∗ = K = { z | Y T z ≽ 0 } (364)