Is the norm of the gradient of a convex function convex (or any convex relaxations of it)?

Question

Given convex set $\mathcal{X}$ and convex function $f : \mathcal{X} \to \mathcal{Y},\; x \mapsto y$ for , is the $p$-norm of the gradient function $\|\nabla f(x)\|_p$ convex in $x$?
Does there exist a closed form convex relaxation for $\|\nabla f(x)\|$ (e.g. dual/convex conjugate)?

I know that $\nabla f(x)$ is monotone in $x$ for convex $f$. But is the norm of a monotone function convex?

EDIT: As pointed out in the counter-example by Arthur, the norm of a gradient in general is not convex.

score 3 · Answer 1 · answered Sep 14 '20 at 15:32

3

Consider $\mathcal X=\mathcal Y=\Bbb R$ and $f(x)=x^4+x$. It is convex, but its derivative is not, even after taking the absolute value.

answered Sep 14 '20 at 15:32

Arthur

My apologies, I goofed. – copper.hat Sep 14 '20 at 16:00
Thanks! A quick followup question then (I'll edit the main question too) --- does there exist some kind of convex relaxation for $| \nabla f(x)|$ (e.g. some closed form dual/convex conjugate)? – Uzair Akbar Sep 14 '20 at 16:19

1 Answers1