$\begingroup$ @Lost1, there are actually four such rules, for each combination of convex/concave inner and outer functions: convex-nondecreasing & convex -> convex, convex-nonincreasing & concave -> convex, concave-nondecreasing & concave -> concave, concave-nonincreasing & convex -> concave. $\endgroup$ –

Aug 18, 2020 · Midpoint-Convex and Continuous Implies Convex 11 Prob. 23, Chap. 4, in Baby Rudin: Every convex function is continuous and every increasing convex function of a convex function is convex

Feb 16, 2015 · Convexity is the exception, not the rule. In my experience, nearly every question "is this function convex?" ends up being answered in the negative---because the cases where convexity is present tend to be somewhat obvious. For general questions of computing derivatives involving vectors and matrices, the Matrix Cookbook is an essential resource.

Nov 18, 2011 · Below is the proof of the fact that every midpoint-convex function is rationally convex, which I copied from my older post on a different forum.

It is easy to prove if you write out (2) based on the definition of convex function. Then what you need to know is that f(.) is convex and the norm is convex. Here any norm is ok, because of any norm is convex. Then the problem is proved.

If the function is twice differentiable and the Hessian is positive semidefinite in the entire domain, then the function is convex. Note that the domain must be assumed to be convex too. If the Hessian has a negative eigenvalue at a point in the interior of the domain, then the function is …

Jun 6, 2017 · I therefore provide here a very general Lemma (with valid reference): Every proper, lower-semi continuous, uniformly convex function on a Banach space is coercive and its subdifferential is onto. Lemma.

Solve a Convex optimization Problem which Involves Non Linear Constraints (Using $ \log \left( \cdot \right) $ Function) 0 Is minimizing the sum of the reciprocals equivalent to maximizing the sum of the non-reciprocals, when the variables are coupled?

Every convex function is continuous. 4. Can a function be neither convex nor concave everywhere? 4.

Aug 23, 2018 · It follows that any log-convex function is also convex, as the exponential of a convex function. In general nothing can be said about the composition of a concave function (like $\log$) with a convex function.