Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ...
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246 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS<br />
3.7.1 monotonic function<br />
A real function <strong>of</strong> real argument is called monotonic when it is exclusively<br />
nonincreasing or nondecreasing over the whole <strong>of</strong> its domain. A real<br />
differentiable function <strong>of</strong> real argument is monotonic when its first derivative<br />
(not necessarily continuous) maintains sign over the function domain.<br />
3.7.1.0.1 Definition. Monotonicity.<br />
Multidimensional function f(X) is<br />
nondecreasing monotonic when<br />
nonincreasing monotonic when<br />
Y ≽ X ⇒ f(Y ) ≽ f(X)<br />
Y ≽ X ⇒ f(Y ) ≼ f(X)<br />
(582)<br />
∀X,Y ∈ dom f .<br />
△<br />
For vector-valued <strong>functions</strong> compared with respect to the nonnegative<br />
orthant, it is necessary and sufficient for each entry f i to be monotonic in<br />
the same sense.<br />
Any affine function is monotonic. tr(Y T X) is a nondecreasing monotonic<br />
function <strong>of</strong> matrix X ∈ S M when constant matrix Y is positive semidefinite,<br />
for example; which follows from a result (356) <strong>of</strong> Fejér.<br />
A <strong>convex</strong> function can be characterized by another kind <strong>of</strong> nondecreasing<br />
monotonicity <strong>of</strong> its gradient:<br />
3.7.1.0.2 Theorem. Gradient monotonicity. [195,B.4.1.4]<br />
[53,3.1 exer.20] Given f(X) : R p×k →R a real differentiable function with<br />
matrix argument on open <strong>convex</strong> domain, the condition<br />
〈∇f(Y ) − ∇f(X) , Y − X〉 ≥ 0 for each and every X,Y ∈ domf (583)<br />
is necessary and sufficient for <strong>convex</strong>ity <strong>of</strong> f . Strict inequality and caveat<br />
distinct Y ,X provide necessary and sufficient conditions for strict <strong>convex</strong>ity.<br />
⋄