What is a concave function divided by a convex function?

Hello all,

I am new to CVX and am excited to start to use it. I apologize if this question is uber noobish and silly.

Basically, I have a function f(x) that is concave. I have another function g(x) that is convex. What I am wondering, is whether the new function h(x) = \frac{f(x)}{g(x)} is convex or concave.

I was wondering if there was a ‘straight forward’ way to show this with some properties and/or rules that I may not know about.

(Actually if you are wondering, the function f(x) is the geometric mean which is concave, and g(x) is the arithmetic mean which is convex). I am wondering what their ratio yields. We also know that \frac{f(x)}{g(x)} is always between 0 and 1. Not sure if that helps but that is the context).

Thanks in advance.

Assuming that the denominator is positive, the ratio of a concave and a convex function is typically quasiconcave. That means you can’t use it in CVX without some change, but in certain models, you can make it work. For instance, f(x)/g(x) \geq \alpha becomes f(x) \geq \alpha g(x), which is valid if \alpha>0.

Quasiconvex and quasiconcave objectives can be handled by solving a sequence of convex feasibility problems. Consult Boyd & Vandenberghe for details.

Hello @mcg thanks I will comment to you on the math.ex because I cant comment on your answer here.

I’m going to delete this here, then, since it’s really not CVX specific.