v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 143
R 2 R 3
1
0.8
(a)
0.6
0.4
0.2
0
∂K ∗
K
∂K ∗
K
(b)
−0.2
−0.4
−0.6
K ∗
−0.8
−1
−0.5 0 0.5 1 1.5
x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ G(K ∗ ) (329)
Figure 49: Dual cone construction by right angle. Each extreme direction of
a proper polyhedral cone is orthogonal to a facet of its dual cone, and vice
versa, in any dimension. (2.13.6.1) (a) This characteristic guides graphical
construction of dual cone in two dimensions: It suggests finding dual-cone
boundary ∂ by making right angles with extreme directions of polyhedral
cone. The construction is then pruned so that each dual boundary vector does
not exceed π/2 radians in angle with each and every vector from polyhedral
cone. Were dual cone in R 2 to narrow, Figure 50 would be reached in limit.
(b) Same polyhedral cone and its dual continued into three dimensions.
(confer Figure 56)