Suppose I have a problem of the form

$mi{n}_{x\ge \u03f5,y\ge 0}\frac{f(x)+g(y)}{x+y}$

subject to some (convex) inequality constraints and some affine equality constraints, and where $f$ and $g$ are known to be convex, and $\u03f5>0$ is some known constant. We can also assume that the feasible set is compact (in addition to being convex).

The main challenge here is when $\frac{f(x)+g(y)}{x+y}$ is not convex (otherwise it is just a convex problem), and also we don't have an easy escape like $f$ and $g$ are log convex, for example.

In my specific situation, I also know that $f,g$ are of the form

$f(x)={\int}_{0}^{x}{h}_{f}(u)\text{}du\text{},$

and

$g(x)={\int}_{0}^{x}{h}_{g}(u)\text{}du\text{},$

for some (known) nondecreasing and integrable (but not necessarily continuous) functions ${h}_{f}:[0,{B}_{f}]\to \mathbb{R}$ and ${h}_{g}:[0,{B}_{g}]\to \mathbb{R}$ for some real numbers $0<{B}_{f},{B}_{g}<\mathrm{\infty}$.

Are there any easy ways to tackle this sort of problem? I am hoping to find ways that would be computationally not much more difficult than convex optimization, but I am not sure if this is possible... it seems like even the simpler problem of minimizing $f(x)/x$ is (surprisingly) difficult...