v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 2052.13.10.1.2 Example. Optimality conditions for conic problem.Consider a convex optimization problem having real differentiable convexobjective function f(x) : R n →R defined on domain R nminimize f(x)xsubject to x ∈ K(454)Let’s first suppose that the feasible set is a pointed polyhedral cone Kpossessing a linearly independent set of generators and whose subspacemembership is made explicit by fat full-rank matrix C ∈ R p×n ; id est, weare given the halfspace-description, for A∈ R m×nK = {x | Ax ≽ 0, Cx = 0} ⊆ R n(287a)(We’ll generalize to any convex cone K shortly.) Vertex-description of thiscone, assuming (ÂZ)† skinny-or-square full-rank, isK = {Z(ÂZ)† b | b ≽ 0} (444)where Â∈ Rm−l×n , l is the number of conically dependent rows in AZ (2.10)which must be removed, and Z ∈ R n×n−rank C holds basis N(C) columnar.From optimality condition (352),because∇f(x ⋆ ) T (Z(ÂZ)† b − x ⋆ )≥ 0 ∀b ≽ 0 (455)−∇f(x ⋆ ) T Z(ÂZ)† (b − b ⋆ )≤ 0 ∀b ≽ 0 (456)x ⋆ Z(ÂZ)† b ⋆ ∈ K (457)From membership relation (450) and Example 2.13.10.1.1〈−(Z T Â T ) † Z T ∇f(x ⋆ ), b − b ⋆ 〉 ≤ 0 for all b ∈ R m−l+⇔(458)−(Z T Â T ) † Z T ∇f(x ⋆ ) ∈ −R m−l+ ∩ b ⋆⊥Then equivalent necessary and sufficient conditions for optimality of conicproblem (454) with feasible set K are: (confer (362))(Z T Â T ) † Z T ∇f(x ⋆ ) ≽R m−l+0, b ⋆ ≽ 0, ∇f(x ⋆ ) T Z(ÂZ)† b ⋆ = 0 (459)R m−l+