v2008.02.02 - Convex Optimization

178 CHAPTER 2. CONVEX GEOMETRYFrom optimality condition (309),because∇f(x ⋆ ) T (Z(ÂZ)† b − x ⋆ )≥ 0 ∀b ≽ 0 (396)−∇f(x ⋆ ) T Z(ÂZ)† (b − b ⋆ )≤ 0 ∀b ≽ 0 (397)x ⋆ ∆ = Z(ÂZ)† b ⋆ ∈ K (398)From membership relation (392) and Example 2.13.10.1.1〈−(Z T Â T ) † Z T ∇f(x ⋆ ), b − b ⋆ 〉 ≤ 0 for all b ∈ R m−l+⇔(399)−(Z T Â T ) † Z T ∇f(x ⋆ ) ∈ −R m−l+ ∩ b ⋆⊥Then the equivalent necessary and sufficient conditions for optimality of theconic program (395) with pointed polyhedral feasible set K are: (confer (315))(Z T Â T ) † Z T ∇f(x ⋆ ) ≽R m−l+0, b ⋆ ≽R m−l+0, ∇f(x ⋆ ) T Z(ÂZ)† b ⋆ = 0 (400)When K = R n + , in particular, then C =0, A=Z =I ∈ S n ; id est,minimize f(x)xsubject to x ≽ 0 (401)The necessary and sufficient conditions become (confer [46,4.2.3])R n +∇f(x ⋆ ) ≽ 0, x ⋆ ≽ 0, ∇f(x ⋆ ) T x ⋆ = 0 (402)R n +R n +2.13.10.1.3 Example. Linear complementarity. [202] [235]Given matrix A ∈ R n×n and vector q ∈ R n , the complementarity problem isa feasibility problem:find w , zsubject to w ≽ 0z ≽ 0w T z = 0w = q + Az(403)