v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 165K is proper if and only if K ∗ is proper.K is polyhedral if and only if K ∗ is polyhedral. [330,2.8]K is simplicial if and only if K ∗ is simplicial. (2.13.9.2) A simplicialcone and its dual are proper polyhedral cones (Figure 64, Figure 54),but not the converse.K ⊞ −K ∗ = R n ⇔ K is closed and convex. (2023)Any direction in a proper cone K is normal to a hyperplane separatingK from −K ∗ .2.13.1.2 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 59)When cone K is a halfspace in R n with n > 0 (Figure 57 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 60), it isneither pointed or full-dimensional; hence, the dual cone K ∗ can be neitherfull-dimensional 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 Lorentz cone (confer (178)){[ ]}xK l = ∈ R n × R | ‖x‖tl ≤ t , l∈{1, ∞} (317)its dual is the proper cone [61, exmp.2.25]{[ ]}K q = K ∗ xl = ∈ R n × R | ‖x‖tq ≤ t , l∈{1, 2, ∞} (318)where ‖x‖ ∗ l = ‖x‖ q is that norm dual to ‖x‖ l determined via solution to1/l + 1/q = 1. Figure 62 illustrates K=K 1 and K ∗ = K ∗ 1= K ∞ in R 2 × R .].