v2009.01.01 - Convex Optimization

228 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
3.1.8.1.4 Exercise. Product and ratio of convex functions. [53, exer.3.32]
In general the product or ratio of two convex functions is not convex. [198]
However, there are some results that apply to functions on R [real domain].
Prove the following. 3.14
(a) If f and g are convex, both nondecreasing (or nonincreasing), and
positive functions on an interval, then fg is convex.
(b) If f , g are concave, positive, with one nondecreasing and the other
nonincreasing, then fg is concave.
(c) If f is convex, nondecreasing, and positive, and g is concave,
nonincreasing, and positive, then f/g is convex.
3.1.9 first-order convexity condition, real function
Discretization of w ≽0 in (444) invites refocus to the real-valued function:
3.1.9.0.1 Theorem. Necessary and sufficient convexity condition.
[53,3.1.3] [110,I.5.2] [342,1.2.3] [37,1.2] [286,4.2] [265,3] 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 (544)
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 (544); for each and every X,Y ∈ domf
f(Y ) ≥ f(X) + ∇f(X) T (Y − X) (545)
From this we can find ∂H − a unique [324,5.5.4] nonvertical [173,B.1.2]
hyperplane [ ](2.4), expressed in terms of function gradient, supporting epif
X
at : videlicet, defining f(Y /∈ domf) = ∆ ∞ [53,3.1.7]
f(X)
[ ] Y
∈ epif ⇔ t ≥ f(Y ) ⇒ [ ∇f(X) T −1 ]([ ] [ ])
Y X
−
≤ 0
t
t f(X)
(546)
3.14 Hint: Prove the result in (a) by verifying Jensen’s inequality.
⋄