v2009.01.01 - Convex Optimization

158 CHAPTER 2. CONVEX GEOMETRY
Via membership relation, we now derive condition (322) from the general
first-order condition for optimality (319): The feasible set for problem (321)
is
C ∆ = {x∈ R n | Cx = d} = {Zξ + x p | ξ ∈ R n−rank C } (323)
where Z ∈ R n×n−rank C holds basis N(C) columnar, and x p is any particular
solution to Cx = d . Since x ⋆ ∈ C , we arbitrarily choose x p = x ⋆ which
yields the equivalent optimality condition
∇f(x ⋆ ) T Zξ ≥ 0 ∀ξ∈ R n−rank C (324)
But this is simply half of a membership relation, and the cone dual to
R n−rank C is the origin in R n−rank C . We must therefore have
Z T ∇f(x ⋆ ) = 0 ⇔ ∇f(x ⋆ ) T Zξ ≥ 0 ∀ξ∈ R n−rank C (325)
meaning, ∇f(x ⋆ ) must be orthogonal to N(C). This condition
Z T ∇f(x ⋆ ) = 0, x ⋆ ∈ C (326)
is necessary and sufficient for optimality of x ⋆ .
2.13.4 Discretization of membership relation
2.13.4.1 Dual halfspace-description
Halfspace-description of the dual cone is equally simple (and extensible
to an infinite number of generators) as vertex-description (94) for the
corresponding closed convex cone: By definition (270), for X ∈ R n×N as in
(253), (confer (259))
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 } (327)
that follows from the generalized inequality and membership corollary (289).
The semi-infinity of tests specified by all z ∈ K has been reduced to a set
of generators for K constituting the columns of X ; id est, the test has been
discretized.