v2009.01.01 - Convex Optimization

3.1. CONVEX FUNCTION 227
For vector-valued functions compared with respect to the nonnegative
orthant, it is necessary and sufficient for each entry f i to be monotonic in
the same sense.
Any affine function is monotonic. tr(Y T X) is a nondecreasing monotonic
function of matrix X ∈ S M when constant matrix Y is positive semidefinite,
for example; which follows from a result (335) of Fejér.
A convex function can be characterized by another kind of nondecreasing
monotonicity of its gradient:
3.1.8.1.2 Theorem. Gradient monotonicity. [173,B.4.1.4]
[48,3.1, exer.20] Given f(X) : R p×k →R a real differentiable function with
matrix argument on open convex domain, the condition
〈∇f(Y ) − ∇f(X) , Y − X〉 ≥ 0 for each and every X,Y ∈ domf (542)
is necessary and sufficient for convexity of f . Strict inequality and caveat
distinct Y ,X provide necessary and sufficient conditions for strict convexity.
⋄
3.1.8.1.3 Example. Composition of functions. [53,3.2.4] [173,B.2.1]
Monotonic functions play a vital role determining convexity of functions
constructed by transformation. Given functions g : R k → R and
h : R n → R k , their composition f = g(h) : R n → R defined by
f(x) = g(h(x)) , dom f = {x∈ domh | h(x)∈ domg} (543)
is convex when
g is convex nondecreasing monotonic and h is convex
g is convex nonincreasing monotonic and h is concave
and composite function f is concave when
g is concave nondecreasing monotonic and h is concave
g is concave nonincreasing monotonic and h is convex
where ∞ (−∞) is assigned to convex (concave) g when evaluated outside
its domain. When functions are differentiable, these rules are consequent to
(1667). Convexity (concavity) of any g is preserved when h is affine.