v2007.09.13 - Convex Optimization

144 CHAPTER 2. CONVEX GEOMETRY2.13.1.2 Examples of dual coneWhen K is R n , K ∗ is the point at the origin, and vice versa.When K is a subspace, K ∗ is its orthogonal complement, and vice versa.(E.9.2.1, Figure 46)When cone K is a halfspace in R n with n > 0 (Figure 44 for example),the dual cone K ∗ is a ray (base 0) belonging to that halfspace but orthogonalto its bounding hyperplane (that contains the origin), and vice versa.When convex cone K is a closed halfplane in R 3 (Figure 47), it is neitherpointed or of nonempty interior; hence, the dual cone K ∗ can be neither ofnonempty interior or pointed.When K is any particular orthant in R n , the dual cone is identical; id est,K = K ∗ .When K is any quadrant in subspace R 2 , K ∗ is a wedge-shaped polyhedralcone in R 3 ; e.g., for K equal to quadrant I , K ∗ =[R2+RWhen K is a polyhedral flavor of the Lorentz cone K l (247), the dual isthe polyhedral proper cone K q : for l=1 or ∞{[ ]}K q = K ∗ xl = ∈ R n × R | ‖x‖tq ≤ t (274)where ‖x‖ q is the dual norm determined via solution to 1/l + 1/q = 1.2.13.2 Abstractions of Farkas’ lemma2.13.2.0.1 Corollary. Generalized inequality and membership relation.[147,A.4.2] Let K be any closed convex cone and K ∗ its dual, and let xand y belong to a vector space R n . Theny ∈ K ∗ ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (275)which is, merely, a statement of fact by definition of dual cone (258). Byclosure we have conjugation:x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ (276)which may be regarded as a simple translation of the Farkas lemma [88] asin [228,22] to the language of convex cones, and a generalization of thewell-known Cartesian factx ≽ 0 ⇔ 〈y , x〉 ≥ 0 for all y ≽ 0 (277)].