v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 185f 1 (x)f 2 (x)(a)(b)Figure 54: 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 forminimizer uniqueness.f convex ⇔ w T f convex ∀w ∈ G(R M + ) (420)shown by substitution of the defining inequality (419). 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} (421)(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 avector-valued function is a comparison on R of each entry.3.1.1.0.1 Exercise. Cone of convex functions.Prove that relation (420) implies: the set of all vector-valued convex functionsin 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 thestrictly convex functions.3.3 The strict case excludes the cone’s point at the origin.