Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.1. CONVEX FUNCTION 213
f 1 (x)
f 2 (x)
(a)
(b)
Figure 66: Convex real functions here have a unique minimizer x ⋆ . For
x∈ R , f 1 (x)=x 2 =‖x‖ 2 2 is strictly convex whereas nondifferentiable function
f 2 (x)= √ x 2 =|x|=‖x‖ 2 is convex but not strictly. Strict convexity of a real
function is only a sufficient condition for minimizer uniqueness.
shown by substitution of the defining inequality (475). Discretization allows
relaxation (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} (477)
(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.
3.1.2 strict convexity
When f(X) instead satisfies, for each and every distinct Y and Z in domf
and all 0