v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 149
K is proper if and only if K ∗ is proper.
K is polyhedral if and only if K ∗ is polyhedral. [286,2.8]
K is simplicial if and only if K ∗ is simplicial. (2.13.9.2) A simplicial
cone and its dual are proper polyhedral cones (Figure 56, Figure 47),
but not the converse.
K ⊞ −K ∗ = R n ⇔ K is closed and convex. (1900) (p.708)
Any direction in a proper cone K is normal to a hyperplane separating
K from −K ∗ .
2.13.1.2 Examples of dual cone
When 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 52)
When cone K is a halfspace in R n with n > 0 (Figure 50 for example),
the dual cone K ∗ is a ray (base 0) belonging to that halfspace but orthogonal
to its bounding hyperplane (that contains the origin), and vice versa.
When convex cone K is a closed halfplane in R 3 (Figure 53), it is neither
pointed or of nonempty interior; hence, the dual cone K ∗ can be neither of
nonempty 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 polyhedral
cone in R 3 ; e.g., for K equal to quadrant I , K ∗ =
[
R
2
+
R
When K is a polyhedral flavor of the Lorentz cone K l (260), the dual is
the proper polyhedral cone K q : for l=1 or ∞
K q = K ∗ l =
{[ x
t
]
.
]
}
∈ R n × R | ‖x‖ q ≤ t
(286)
where ‖x‖ q is the dual norm determined via solution to 1/l + 1/q = 1.