v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 231
f(Y ) − f(X) −
→Y −X
df(X) ≽
0 ⇔
〈
〉
→Y −X
f(Y ) − f(X) − df(X) , w ≥ 0 ∀w ≽ 0
where
R M +
→Y −X
df(X) =
⎡
⎢
⎣
(552)
tr ( ∇f 1 (X) T (Y − X) ) ⎤
tr ( ∇f 2 (X) T (Y − X) )
⎥
.
tr ( ∇f M (X) T (Y − X) ) ⎦ ∈ RM (553)
Necessary and sufficient discretization (444) allows relaxation of the
semi-infinite number of conditions w ≽ 0 instead to w ∈ {e i , i=1... M}
the extreme directions of the nonnegative orthant. Each extreme direction
→Y −X
picks out a real entry f i and df(X) i
from vector-valued function f and its
→Y −X
directional derivative df(X) , then Theorem 3.1.9.0.1 applies.
The vector-valued function case (551) is therefore a straightforward
application of the first-order convexity condition for real functions to each
entry of the vector-valued function.
R M +
3.1.11 second-order convexity condition
Again, by discretization (444), 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 (554)
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 (554) demands nonnegative curvature, intuitively, hence
precluding points of inflection as in Figure 69 (p.240).
Strict inequality is 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 is reliably true.