v2010.10.26 - Convex Optimization

3.1. CONVEX FUNCTION 221f 1 (x)f 2 (x)(a)(b)Figure 68: Convex real functions here have a unique minimizer x ⋆ . Forx∈ R , f 1 (x)=x 2 =‖x‖ 2 2 is strictly convex whereas nondifferentiable functionf 2 (x)= √ x 2 =|x|=‖x‖ 2 is convex but not strictly. Strict convexity of a realfunction is only a sufficient condition for minimizer uniqueness.shown by substitution of the defining inequality (497). Discretization allowsrelaxation (2.13.4.2.1) of a semiinfinite number of conditions {w ∈ R M∗+ } to:{w ∈ G(R M∗+ )} ≡ {e i ∈ R M , i=1... M } (499)(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.3.1.2 strict convexityWhen f(X) instead satisfies, for each and every distinct Y and Z in domfand all 0