v2009.01.01 - Convex Optimization

144 CHAPTER 2. CONVEX GEOMETRY
K
K ∗
0
Figure 50: K is a halfspace about the origin in R 2 . K ∗ is a ray base 0,
hence has empty interior in R 2 ; so K cannot be pointed. (Both convex
cones appear truncated.)
2.13.1.0.2 Exercise. Dual cone definitions.
What is {x∈ R n | x T z ≥0 ∀z∈R n } ?
What is {x∈ R n | x T z ≥1 ∀z∈R n } ?
What is {x∈ R n | x T z ≥1 ∀z∈R n +} ?
As defined, dual cone K ∗ exists even when the affine hull of the original
cone is a proper subspace; id est, even when the original cone has empty
interior. Rockafellar formulates the dimension of K and K ∗ . [266,14] 2.52
To further motivate our understanding of the dual cone, consider the
ease with which convergence can be observed in the following optimization
problem (p):
2.13.1.0.3 Example. Dual problem. (confer4.1)
Duality is a powerful and widely employed tool in applied mathematics for
a number of reasons. First, the dual program is always convex even if the
primal is not. Second, the number of variables in the dual is equal to the
number of constraints in the primal which is often less than the number of
2.52 His monumental work Convex Analysis has not one figure or illustration. See
[25,II.16] for a good illustration of Rockafellar’s recession cone [37].