Minimizing $\sum_{i=1}^n x_i/y_i$ over a polytope
Your objective function is indeed not convex, but you can make it convex with the appropriate transformation, which means it is solvable in polynomial time. But applying this reformulation is complex: you should try a non-convex solver first and stop there if it works in practice.

The easy way: Non-convex quadratic model

As pointed in the comments, it is a quadratic program, so you can solve it with a non-convex quadratic solver. Although not a polynomial-time algorithm theoretically, these solvers are really efficient in practice. 

\begin{align*} {\rm minimize}&amp;~ \sum_{i=1}^n r_i \\
\mbox{ subject to } &amp; \\
&amp; Ax + B y \leq c \\ 
&amp; r_iy_i \geq x_i \\
&amp; x&gt;0, y&gt;0
\end{align*}

The hard way: Convex fractional programming reformulation

Stated as follows, it is clear that it is a fractional programming problem:
\begin{align*} {\rm minimize}&amp;~ \frac{\sum_{i=1}^n x_i \prod_{j\neq i} y_j}{\prod_{j=1}^n y_j} \\
\mbox{ subject to }&amp; \\
&amp; Ax + B y \leq c \\ 
&amp; r_i \geq x_iy_i \\
&amp; x&gt;0, y&gt;0
\end{align*}

Both the numerator and denominator are convex, so you can follow the transformation given in the above link to make it convex. This makes it solvable in polynomial time.

$$
\begin{align*}
z_i &amp;= \frac{x_i}{\prod_{j=1}^n y_j} \\
w_i &amp;= \frac{y_i}{\prod_{j=1}^n y_j} \\
t &amp;= \frac{1}{\prod_{j=1}^n y_j} \\
\end{align*}
$$

And solve (yes, the convexity is far from obvious):

$$
\begin{align*} {\rm minimize}&amp; ~ \frac{1}{t^{n-1}} \sum_{i=1}^n z_i \prod_{j\neq i} w_j \\
\mbox{ subject to } &amp; \\
&amp; \frac{\prod_{j\neq i} w_j}{t^{n-1}} \leq 1 \\
&amp; Ax + By \leq ct \\ 
&amp; z&gt;0, w&gt;0, t \geq 0
\end{align*}
$$