Is a set of convex functions closed under composition? I don’t necessarily need a proof, but a reference would be greatly appreciated.
There is no need for the first function in the composition to be nondecreasing. And here is a proof for the nondifferentiable case as well. The only assumptions are that the composition is well defined at the points involved in the proof for every α∈[0,1] and that fn,fn−1,…,f1 are convex nondecreasing functions of one variable and that f0:Rn→R is a convex function.
First let g:Rm→R a convex function and f:R→R a convex nondecreasing function, then, by convexity of g:
So, using the fact that f is nondecreasing:
Therefore, again by convexity:
This reasoning can be used inductively in order to prove the result that
is convex under the stated hypothesis. And the composition will be nondecreasing if f0 is nondecreasing.