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

252 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
Necessary and sufficient discretization (476) allows relaxation of the
semiinfinite number of conditions {w ≽ 0} instead to {w ∈ {e i , i=1... M }}
the extreme directions of the selfdual nonnegative orthant. Each extreme
→Y −X
direction picks out a real entry f i and df(X) i
from the vector-valued
function and its directional derivative, then Theorem 3.7.2.0.1 applies.
The vector-valued function case (592) is therefore a straightforward
application of the first-order convexity condition for real functions to each
entry of the vector-valued function.
3.7.4 second-order convexity condition
Again, by discretization (476), we are obliged only to consider each individual
entry f i of a vector-valued function f ; id est, the real functions {f i }.
For f(X) : R p →R M , a twice differentiable vector-valued function with
vector argument on open convex domain,
∇ 2 f i (X) ≽
0 ∀X ∈ domf , i=1... M (595)
S p +
is a necessary and sufficient condition for convexity of f . Obviously,
when M = 1, this convexity condition also serves for a real function.
Condition (595) demands nonnegative curvature, intuitively, hence
precluding points of inflection as in Figure 77 (p.260).
Strict inequality in (595) provides only a sufficient condition for strict
convexity, but that is nothing new; videlicet, the strictly convex real function
f i (x)=x 4 does not have positive second derivative at each and every x∈ R .
Quadratic forms constitute a notable exception where the strict-case converse
holds reliably.
3.7.4.0.1 Exercise. Real fractional function. (confer3.4,3.6.0.0.4)
Prove that real function f(x,y) = x/y is not convex on the first quadrant.
Also exhibit this in a plot of the function. (f is quasilinear (p.261) on
{y > 0} , in fact, and nonmonotonic even on the first quadrant.)
3.7.4.0.2 Exercise. Stress function.
Define |x −y| √ (x −y) 2 and
X = [x 1 · · · x N ] ∈ R 1×N (76)