v2007.09.13 - Convex Optimization

3.1. CONVEX FUNCTION 2093.1.8.1 monotonic functionA real differentiable function f of real argument is called monotonic when itsfirst derivative (not necessarily continuous) maintains sign over the functiondomain.3.1.8.1.1 Definition. Monotonicity.Multidimensional function f is monotonic when sgn〈f(Y )−f(X), Y −X〉is invariant (ignoring 0) to all X,Y ∈ dom f . Nonnegative (nonpositive)sign denotes nonnegative (nonpositive) monotonicity.△When argument X and f(X) are dimensionally incompatible, the onehaving smaller dimension is padded with 1 to complete the test. It isnecessary and sufficient for each entry f i from this monotonicity definitionto be monotonic with the same sign.A convex function can be characterized by a similar kind of nonnegativemonotonicity of its gradient:3.1.8.1.2 Theorem. Gradient monotonicity. [147,B.4.1.4][41,3.1, exer.20] Given f(X) : R p×k →R a real differentiable function withmatrix argument on open convex domain, the condition〈∇f(Y ) − ∇f(X) , Y − X〉 ≥ 0 for each and every X,Y ∈ domf (507)is necessary and sufficient for convexity of f . Strict inequality and caveatdistinct Y ,X provide necessary and sufficient conditions for strict convexity.⋄3.1.8.1.3 Example. Composition of functions. [46,3.2.4] [147,B.2.1]Monotonic functions play a vital role determining convexity of functionsconstructed by transformation. Given functions g : R k → R andh : R n → R k , their composition f = g(h) : R n → R defined byf(x) = g(h(x)) , dom f = {x∈ domh | h(x)∈ domg} (508)is convex when