v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 141
2.13.1 Dual cone
For any set K (convex or not), the dual cone [92,4.2]
K ∗ ∆ = { y ∈ R n | 〈y , x〉 ≥ 0 for all x ∈ K } (270)
is a unique cone 2.51 that is always closed and convex because it is an
intersection of halfspaces (halfspaces theorem (2.4.1.1.1)) whose partial
boundaries each contain the origin, each halfspace having inward-normal x
belonging to K ; e.g., Figure 48(a).
When cone K is convex, there is a second and equivalent construction:
Dual cone K ∗ is the union of each and every vector y inward-normal to
a hyperplane supporting K or bounding a halfspace containing K ; e.g.,
Figure 48(b). When K is represented by a halfspace-description such as
(259), for example, where
⎡
a T1
A =
∆ ⎣ .
⎤
⎡
c T1
⎦∈ R m×n , C =
∆ ⎣ .
⎤
⎦∈ R p×n (271)
a T m
c T p
then the dual cone can be represented as the conic hull
K ∗ = cone{a 1 ,..., a m , ±c 1 ,..., ±c p } (272)
a vertex-description, because each and every conic combination of normals
from the halfspace-description of K yields another inward-normal to a
hyperplane supporting or bounding a halfspace containing K .
K ∗ can also be constructed pointwise using projection theory fromE.9.2:
for P K x the Euclidean projection of point x on closed convex cone K
−K ∗ = {x − P K x | x∈ R n } = {x∈ R n | P K x = 0} (1901)
2.13.1.0.1 Exercise. Dual cone manual-construction.
Perhaps the most instructive graphical method of dual cone construction is
cut-and-try. Find the dual of each polyhedral cone from Figure 49 by using
dual cone equation (270).
2.51 The dual cone is the negative polar cone defined by many authors; K ∗ = −K ◦ .
[173,A.3.2] [266,14] [36] [25] [286,2.7]