v2010.10.26 - Convex Optimization

206 CHAPTER 2. CONVEX GEOMETRYexpressible, by (445),∇f(x ⋆ ) ∈ K ∗ , x ⋆ ∈ K , ∇f(x ⋆ ) T x ⋆ = 0 (460)This result (460) actually applies more generally to any convex cone Kcomprising the feasible set: Necessary and sufficient optimality conditionsare in terms of objective gradient−∇f(x ⋆ ) ∈ −(K − x ⋆ ) ∗ , x ⋆ ∈ K (452)whose membership to normal cone, assuming only cone K convexity,−(K − x ⋆ ) ∗ = K ⊥ K(x ⋆ ∈ K) = −K ∗ ∩ x ⋆⊥ (2091)equivalently expresses conditions (460).When K = R n + , in particular, then C =0, A=Z =I ∈ S n ; id est,minimize f(x)xsubject to x ≽ 0 (461)Necessary and sufficient optimality conditions become (confer [61,4.2.3])R n +∇f(x ⋆ ) ≽ 0, x ⋆ ≽ 0, ∇f(x ⋆ ) T x ⋆ = 0 (462)R n +R n +equivalent to condition (330) 2.81 (under nonzero gradient) for membershipto the nonnegative orthant boundary ∂R n + .2.13.10.1.3 Example. Complementarity problem. [210]A complementarity problem in nonlinear function f is nonconvexfind z ∈ Ksubject to f(z) ∈ K ∗〈z , f(z)〉 = 0(463)yet bears strong resemblance to (460) and to (2027) Moreau’s decompositiontheorem on page 740 for projection P on mutually polar cones K and −K ∗ .2.81 and equivalent to well-known Karush-Kuhn-Tucker (KKT) optimality conditions[61,5.5.3] because the dual variable becomes gradient ∇f(x).