v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 2032.13.10 Dual cone-translate(E.10.3.2.1) First-order optimality condition (352) inspires a dual-conevariant: For any set K , the negative dual of its translation by any a∈ R n is−(K − a) ∗ = {y ∈ R n | 〈y , x − a〉≤0 for all x ∈ K} K ⊥ (a)= {y ∈ R n | 〈y , x〉≤0 for all x ∈ K − a}(449)a closed convex cone called normal cone to K at point a . From this, a newmembership relation like (320):y ∈ −(K − a) ∗ ⇔ 〈y , x − a〉≤0 for all x ∈ K (450)and by closure the conjugate, for closed convex cone Kx ∈ K ⇔ 〈y , x − a〉≤0 for all y ∈ −(K − a) ∗ (451)2.13.10.1 first-order optimality condition - restatement(confer2.13.3) The general first-order necessary and sufficient condition foroptimality of solution x ⋆ to a minimization problem with real differentiableconvex objective function f(x) : R n →R over convex feasible set C is [306,3]−∇f(x ⋆ ) ∈ −(C − x ⋆ ) ∗ , x ⋆ ∈ C (452)id est, the negative gradient (3.6) belongs to the normal cone to C at x ⋆ asin Figure 67.2.13.10.1.1 Example. Normal cone to orthant.Consider proper cone K = R n + , the selfdual nonnegative orthant in R n . Thenormal cone to R n + at a∈ K is (2091)K ⊥ R n + (a∈ Rn +) = −(R n + − a) ∗ = −R n + ∩ a ⊥ , a∈ R n + (453)where −R n += −K ∗ is the algebraic complement of R n + , and a ⊥ is theorthogonal complement to range of vector a . This means: When point ais interior to R n + , the normal cone is the origin. If n p represents numberof nonzero entries in vector a∈∂R n + , then dim(−R n + ∩ a ⊥ )= n − n p andthere is a complementary relationship between the nonzero entries in vector aand the nonzero entries in any vector x∈−R n + ∩ a ⊥ .