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

3.7. GRADIENT 251
meaning, the gradient at X identifies a supporting hyperplane there in R p
{Y ∈ R p | ∇f(X) T (Y − X) = 0} (590)
to the convex sublevel sets of convex function f (confer (537))
L f(X) f {Z ∈ domf | f(Z) ≤ f(X)} ⊆ R p (591)
illustrated for an arbitrary real convex function in Figure 76 and Figure 65.
That supporting hyperplane is unique for twice differentiable f . [214, p.501]
3.7.3 first-order convexity condition, vector function
Now consider the first-order necessary and sufficient condition for convexity of
a vector-valued function: Differentiable function f(X) : R p×k →R M is convex
if and only if domf is open, convex, and for each and every X,Y ∈ domf
f(Y ) ≽
R M +
f(X) +
→Y −X
df(X)
= f(X) + d dt∣ f(X+ t (Y − X)) (592)
t=0
→Y −X
where df(X) is the directional derivative 3.19 [214] [326] of f at X in direction
Y −X . This, of course, follows from the real-valued function case: by dual
generalized inequalities (2.13.2.0.1),
f(Y ) − f(X) −
where
→Y −X
df(X) ≽
R M +
→Y −X
df(X) =
0 ⇔
⎡
⎢
⎣
〈
〉
→Y −X
f(Y ) − f(X) − df(X) , w ≥ 0 ∀w ≽ 0
R M∗
+
(593)
tr ( ∇f 1 (X) T (Y − X) ) ⎤
tr ( ∇f 2 (X) T (Y − X) )
⎥
.
tr ( ∇f M (X) T (Y − X) ) ⎦ ∈ RM (594)
3.19 We extend the traditional definition of directional derivative inD.1.4 so that direction
may be indicated by a vector or a matrix, thereby broadening the scope of the Taylor
series (D.1.7). The right-hand side of the inequality (592) is the first-order Taylor series
expansion of f about X .