Maximization of sum of linear fractional functions

Question

The following optimization problem in terms of maximizing the sum of fractional functions is given:

\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & \sum_{i=1}^{p} \frac{1}{b_i-a_i^Tx} \\ & \text{subject to} & & x\in X, \end{aligned} \end{equation*}

where $b_i-a_i^Tx>0$ and $X$ is a convex set. How can one solve this optimization problem?

Note that the corresponding minimization problem has already been solved in minimization-of-sum-of-linear-fractional-functions by expressing the objective function in SOC form, but as far as I can tell that solution method is not directly applicable to the problem at hand in this thread.

Answer 1

For both minimization and maximization, you can apply a Charnes-Cooper transformation (with a separate $t_i$ for each summand) to make the objective linear. In the special case that $X$ is a polyhedron, this transformation yields a linear programming problem.