v2010.10.26 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 159R 2 R 310.8(a)0.60.40.20∂K ∗K∂K ∗K(b)−0.2−0.4−0.6K ∗−0.8−1−0.5 0 0.5 1 1.5x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) (366)Figure 56: Dual cone construction by right angle. Each extreme direction ofa proper polyhedral cone is orthogonal to a facet of its dual cone, and viceversa, in any dimension. (2.13.6.1) (a) This characteristic guides graphicalconstruction of dual cone in two dimensions: It suggests finding dual-coneboundary ∂ by making right angles with extreme directions of polyhedralcone. The construction is then pruned so that each dual boundary vector doesnot exceed π/2 radians in angle with each and every vector from polyhedralcone. Were dual cone in R 2 to narrow, Figure 57 would be reached in limit.(b) Same polyhedral cone and its dual continued into three dimensions.(confer Figure 64)