positive constant divided by a concave function, how to convexify this constraint?

Question

I have following constraint in my optimization problem:

$0 \leq t \leq \frac{S}{B \ R(p)}$

where $t,p$ are optimization variables and $S,B$ are positive constants. $R(p)$ is a concave function which is in the denominator, is there any way to make this constraint convex?

Thanks in advance!

Answer 1

From your constraint, it's clear $R$ is positive as well. Then,we have $$\frac1{R(p)}=\max_{\lambda>0}\left(2\lambda-\lambda^2R(p)\right)$$ and since $R$ is concave, that's the maximum of a set of convex functions in $p$, i.e. it's convex (for a proof, look here).