v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 197
f 1 (x)
f 2 (x)
(a)
(b)
Figure 60: Each convex real function has a unique minimizer x ⋆ but,
for x∈ R , f 1 (x)=x 2 is strictly convex whereas f 2 (x)= √ x 2 =|x| is not.
Strict convexity of a real function is therefore only a sufficient condition for
minimizer uniqueness.
f convex ⇔ w T f convex ∀w ∈ G(R M + ) (444)
shown by substitution of the defining inequality (443). Discretization
(2.13.4.2.1) allows relaxation of the semi-infinite number of conditions
∀w ≽ 0 to:
∀w ∈ G(R M + ) = {e i , i=1... M} (445)
(the standard basis for R M and a minimal set of generators (2.8.1.2) for R M + )
from which the stated conclusion follows; id est, the test for convexity of a
vector-valued function is a comparison on R of each entry.
3.1.1.0.1 Exercise. Cone of convex functions.
Prove that relation (444) implies: the set of all vector-valued convex functions
in R M is a convex cone. Indeed, any nonnegatively weighted sum of (strictly)
convex functions remains (strictly) convex. 3.3 Interior to the cone are the
strictly convex functions:
3.3 Strict case excludes cone’s point at origin. By these definitions (444) (447), positively
weighted sums mixing convex and strictly convex real functions are not strictly convex
because each summand is considered to be an entry from a vector-valued function.