Consider the LP (linear programming) $$\min \frac{\max_{i=1,..,m}a_i^Tx+b_i}{\min_{j=1,..,p}c_j^Tx+d_j} \\ s.t \qquad Fx \leq g$$ assume that $c_j^Tx+d_j > 0$ for all $x$ satisfying $ Fx \leq g$.
How can I reformulate above problem as a linear programming. If $m=1, p=1$, it is not hard by put $y=x/(c^Tx+d), z = 1/(c^Tx+d)$, then we get min $a^Ty+bz$ subject to $Fy-gz \leq 0$. But I don't know how to generalize it. Can anyone give me some hints?
UPDATE: If someone is interested in the solution, I have an answer.
Put $y:=\frac{x}{\min_{j=1,..,p}c_j^Tx+d_j}, z:= \frac{1}{\min_{j=1,..,p}c_j^Tx+d_j}$. Hence, $x = \frac yz$ and the problem can be rewritten as $$\min {\max_{i=1,..,m}a_i^Ty+b_iz} \\ s.t \qquad Fy \leq gz$$ and we have more contraints $c_j^Ty+d_jz =\frac{c_j^Tx+d_j}{\min_{j=1,..,p}c_j^Tx+d_j} \geq 1$ for all $j = 1,\ldots, p$. Put them together we can reformulate the problem as $$\min \quad t \\ s.t\qquad a_i^Ty+b_iz \leq t \quad \forall i \\ \qquad Fy \leq gz\\ c_j^Ty+d_jz \geq 1 \forall j$$
Rewrite the problem to $\min \{ t_1 / t_2 : t_1 \geq a_i^Tx + b_i, t_2 \leq c_j^T x + d_j \}$. This is a fractional optimization problem (fractional objective and linear constraints), so now you can apply the reformulation you wanted to apply to the original problem.