Let $S_1,S_2\subseteq [N]$.
I am dealing with an optimization problem that is compliant with the GP formulation except for a set of inequalities constraints of the type:
$$ \left(\prod_{i \in S_1} x_i \left(\sum_{j\in S_1^c} y_j\right)\right)^{-1} + \left(\prod_{k\in S_2} x_k \left(\sum_{l \in S_2^c} y_l\right)\right)^{-1} \le 1, $$ where the summations and products are taken over different sets, and the objective $$ \min_{x,y} a_{i,j}\left(\prod_{i\in S_1} x_i \left(\sum_{j\in S_1^c} y_j\right)\right) + a_{k,l}\left(\prod_{k\in S_2} x_k \left(\sum_{l\in S_2^c} y_l\right)\right), $$ where $a\succeq 0$. Obviously, the constraint is not in posynomial form since it contains reciprocals of posynomials.
I was hoping to apply the trick explained in Sec. 5.1 to transform these inequalities as
$$ \left(\sum_{j\in S_1^c} y_j\right) \le t_j\ \\ \left(\sum_{l\in S_2^c} y_l\right) \le t_l\ \\ \left(t_j\prod_{i\in S_1} x_i \right)^{-1} + \left(t_l\prod_{k\in S_2} x_k \right)^{-1} \le 1, $$ where $t_i, t_j$ are positive auxiliary variables. The method in Sec. 5.1 is however presented for fractional positive powers of posynomials.
In the reference, they explain that if the $t$ are increasing functions of the $f(y)$ in the constraints (in this case $\sum_{j}y_j$ for example) then the constraints should be equivalent. Can someone provide an explanation of why this holds or prove that this approach doesn't work for reciprocals of posynomials?