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

248 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.7.2 first-order convexity condition, real function
Discretization of w ≽0 in (476) invites refocus to the real-valued function:
3.7.2.0.1 Theorem. Necessary and sufficient convexity condition.
[59,3.1.3] [129,I.5.2] [384,1.2.3] [41,1.2] [324,4.2] [300,3] For real
differentiable function f(X) : R p×k →R with matrix argument on open
convex domain, the condition (conferD.1.7)
f(Y ) ≥ f(X) + 〈∇f(X) , Y − X〉 for each and every X,Y ∈ domf (585)
is necessary and sufficient for convexity of f .
⋄
When f(X) : R p →R is a real differentiable convex function with vector
argument on open convex domain, there is simplification of the first-order
condition (585); for each and every X,Y ∈ domf
f(Y ) ≥ f(X) + ∇f(X) T (Y − X) (586)
From this we can find ∂H − a unique [364, p.220-229] nonvertical [195,B.1.2]
hyperplane [ ](2.4), expressed in terms of function gradient, supporting epif
X
at : videlicet, defining f(Y /∈ domf) ∞ [59,3.1.7]
f(X)
[ ] Y
∈ epif ⇔ t ≥ f(Y ) ⇒ [ ∇f(X) T −1 ]([ ] [ ])
Y X
− ≤ 0
t
t f(X)
(587)
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 (588)
t R
t f(X)
One such supporting hyperplane (confer Figure 29a) is illustrated in
Figure 75 for a convex quadratic.
From (586) we deduce, for each and every X,Y ∈ domf
∇f(X) T (Y − X) ≥ 0 ⇒ f(Y ) ≥ f(X) (589)