Chapter 3 Geometry of convex functions - Meboo Publishing ...
Chapter 3 Geometry of convex functions - Meboo Publishing ... Chapter 3 Geometry of convex functions - Meboo Publishing ...
246 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS 3.7.1 monotonic function A real function of real argument is called monotonic when it is exclusively nonincreasing or nondecreasing over the whole of its domain. A real differentiable function of real argument is monotonic when its first derivative (not necessarily continuous) maintains sign over the function domain. 3.7.1.0.1 Definition. Monotonicity. Multidimensional function f(X) is nondecreasing monotonic when nonincreasing monotonic when Y ≽ X ⇒ f(Y ) ≽ f(X) Y ≽ X ⇒ f(Y ) ≼ f(X) (582) ∀X,Y ∈ dom f . △ 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 (356) of Fejér. A convex function can be characterized by another kind of nondecreasing monotonicity of its gradient: 3.7.1.0.2 Theorem. Gradient monotonicity. [195,B.4.1.4] [53,3.1 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 (583) is necessary and sufficient for convexity of f . Strict inequality and caveat distinct Y ,X provide necessary and sufficient conditions for strict convexity. ⋄
3.7. GRADIENT 247 3.7.1.0.3 Example. Composition of functions. [59,3.2.4] [195,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)) , domf = {x∈ dom h | h(x)∈ domg} (584) 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. For differentiable functions, these rules are consequent to (1750). Convexity (concavity) of any g is preserved when h is affine. A squared norm is convex having the same minimum because a squaring operation is convex nondecreasing monotonic on the nonnegative real line. 3.7.1.0.4 Exercise. Product and ratio of convex functions. [59, exer.3.32] In general the product or ratio of two convex functions is not convex. [225] However, there are some results that apply to functions on R [real domain]. Prove the following. 3.18 (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. 3.18 Hint: Prove3.7.1.0.4a by verifying Jensen’s inequality ((475) at µ= 1 2 ).
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3.7. GRADIENT 247<br />
3.7.1.0.3 Example. Composition <strong>of</strong> <strong>functions</strong>. [59,3.2.4] [195,B.2.1]<br />
Monotonic <strong>functions</strong> play a vital role determining <strong>convex</strong>ity <strong>of</strong> <strong>functions</strong><br />
constructed by transformation. Given <strong>functions</strong> g : R k → R and<br />
h : R n → R k , their composition f = g(h) : R n → R defined by<br />
f(x) = g(h(x)) , domf = {x∈ dom h | h(x)∈ domg} (584)<br />
is <strong>convex</strong> when<br />
g is <strong>convex</strong> nondecreasing monotonic and h is <strong>convex</strong><br />
g is <strong>convex</strong> nonincreasing monotonic and h is concave<br />
and composite function f is concave when<br />
g is concave nondecreasing monotonic and h is concave<br />
g is concave nonincreasing monotonic and h is <strong>convex</strong><br />
where ∞ (−∞) is assigned to <strong>convex</strong> (concave) g when evaluated outside its<br />
domain. For differentiable <strong>functions</strong>, these rules are consequent to (1750).<br />
Convexity (concavity) <strong>of</strong> any g is preserved when h is affine. <br />
A squared norm is <strong>convex</strong> having the same minimum because a squaring<br />
operation is <strong>convex</strong> nondecreasing monotonic on the nonnegative real line.<br />
3.7.1.0.4 Exercise. Product and ratio <strong>of</strong> <strong>convex</strong> <strong>functions</strong>. [59, exer.3.32]<br />
In general the product or ratio <strong>of</strong> two <strong>convex</strong> <strong>functions</strong> is not <strong>convex</strong>. [225]<br />
However, there are some results that apply to <strong>functions</strong> on R [real domain].<br />
Prove the following. 3.18<br />
(a) If f and g are <strong>convex</strong>, both nondecreasing (or nonincreasing), and<br />
positive <strong>functions</strong> on an interval, then fg is <strong>convex</strong>.<br />
(b) If f , g are concave, positive, with one nondecreasing and the other<br />
nonincreasing, then fg is concave.<br />
(c) If f is <strong>convex</strong>, nondecreasing, and positive, and g is concave,<br />
nonincreasing, and positive, then f/g is <strong>convex</strong>.<br />
<br />
3.18 Hint: Prove3.7.1.0.4a by verifying Jensen’s inequality ((475) at µ= 1 2 ).
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