v2009.01.01 - Convex Optimization

184 CHAPTER 2. CONVEX GEOMETRY
2.13.10.1.2 Example. Optimality conditions for conic problem.
Consider a convex optimization problem having real differentiable convex
objective function f(x) : R n →R defined on domain R n
minimize f(x)
x
subject to x ∈ K
(407)
The feasible set is a pointed polyhedral cone K possessing a linearly
independent set of generators and whose subspace membership is made
explicit by fat full-rank matrix C ∈ R p×n ; id est, we are given the
halfspace-description
K = {x | Ax ≽ 0, Cx = 0} ⊆ R n
(259a)
where A∈ R m×n . The vertex-description of this cone, assuming (ÂZ)†
skinny-or-square full-rank, is
K = {Z(ÂZ)† b | b ≽ 0} (398)
where Â∈ Rm−l×n , l is the number of conically dependent rows in AZ
(2.10) that must be removed, and Z ∈ R n×n−rank C holds basis N(C)
columnar.
From optimality condition (319),
because
∇f(x ⋆ ) T (Z(ÂZ)† b − x ⋆ )≥ 0 ∀b ≽ 0 (408)
−∇f(x ⋆ ) T Z(ÂZ)† (b − b ⋆ )≤ 0 ∀b ≽ 0 (409)
x ⋆ ∆ = Z(ÂZ)† b ⋆ ∈ K (410)
From membership relation (404) and Example 2.13.10.1.1
〈−(Z T Â T ) † Z T ∇f(x ⋆ ), b − b ⋆ 〉 ≤ 0 for all b ∈ R m−l
+
⇔
(411)
−(Z T Â T ) † Z T ∇f(x ⋆ ) ∈ −R m−l
+ ∩ b ⋆⊥
Then the equivalent necessary and sufficient conditions for optimality of the
conic problem (407) with pointed polyhedral feasible set K are: (confer (326))