Suppose I have a problem of the form $$min_{x \geq \epsilon, y \geq 0} \frac{f(x) + g(y)}{x+y}$$ subject to some (convex) inequality constraints and some affine equality constraints, and where $f$ and $g$ are known to be convex, and $\epsilon > 0$ 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
$$f(x) = \int_0^x h_f(u)\ du~,$$ and $$g(x) = \int_0^x h_g(u)\ du~,$$ for some (known) nondecreasing and integrable (but not necessarily continuous) functions $h_f: [0,B_f] \rightarrow \mathbb R$ and $h_g: [0,B_g] \rightarrow \mathbb R$ for some real numbers $0 < B_f, B_g < \infty$.
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...
You can solve it by bisection, i.e. use bisection to find smallest possible $t$ such that $f(x) + g(x) \leq t(x+y)$. For fixed $t$ this is a convex set, and you would thus use your convex programming machinery if available for your particular functions.