v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 229
This means, for each and every point X in the domain of a real convex
[ function f(X) ] , there exists a hyperplane ∂H − [ in R p × ] R having normal
∇f(X)
X
supporting the function epigraph at ∈ ∂H
−1
f(X)
−
{[ ] [ ] Y R
p [
∂H − = ∈ ∇f(X) T −1 ]([ ] [ ]) }
Y X
− = 0
t R
t f(X)
(547)
One such supporting hyperplane (confer Figure 25(a)) is illustrated in
Figure 67 for a convex quadratic.
From (545) we deduce, for each and every X,Y ∈ domf
∇f(X) T (Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (548)
meaning, the gradient at X identifies a supporting hyperplane there in R p
{Y ∈ R p | ∇f(X) T (Y − X) = 0} (549)
to the convex sublevel sets of convex function f (confer (496))
L f(X) f ∆ = {Y ∈ domf | f(Y ) ≤ f(X)} ⊆ R p (550)
illustrated for an arbitrary real convex function in Figure 68.
3.1.10 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)) (551)
t=0
→Y −X
where df(X) is the directional derivative 3.15 [190] [288] 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),
3.15 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 (551) is the first-order Taylor series
expansion of f about X .