I want to solve a linear programming problem in the $2n$ variables $x_1, \ldots, x_n$ and $y_1, \ldots, y_n$ where the cost is of the form $x_1/y_1 + \cdots + x_n/y_n$. Specifically, I want to solve \begin{align*} & {\rm minimize}~ \sum_{i=1}^n x_i/y_i \\ &\mbox{ subject to } Ax + B y \leq c \\ & x>0, y>0, \end{align*} where $A$ and $B$ are given matrices and $c$ is a given vector.
This feels like it should be NP-hard due to the non-convexity of the objective, but I have no idea how to show this, or even whether I'm missing some kind of clever trick that makes it convex. If the matrices $A$ and $B$ were positive, you could plug in $x_i=e^{u_i}, y_i = e^{v_i}$ and the problem would become a convex problem that you could solve via geometric programming, but positivity is not an assumption. I'm not even sure what would be a natural combinatorial problem to try to write in this format, so any suggestions would be very welcome.