Solving a LFP-Problem with as Simplex subprocedure

Question

I'm a computer science student, who is trying to solve a Linear-fractional programming (LFP) problem for an exercise in a course about Decision Support Systems.

In the said exercise I need to provide an algorithm, calling a Simplex implementation as subprocedure, for solving the following general problem:

$$min \quad \frac{cx+d}{fx+g}$$ $$s.t. \quad Ax \leq b, \quad fx + g > 0$$ $$with \quad c, f \in \mathbb{R}^n, \quad d, g \in \mathbb{R}, \quad A \in \mathbb{R}^{m \times n}, \quad b \in \mathbb{R}^m $$

I figured out that I can transform the LFP into a LP by applying the Charnes-Cooper Transformation and then apply Simplex to the result of that. My questions now are:

1. Is the Charnes-Cooper Transformation the right way to go?
2. Since I can only find sources with the Charnes-Cooper Transformation being applied to maximization problems, does the transformation process change when minimizing?
3. How do I turn the result of the transformation into the form Simplex can work with? It seems to me like the algorithm would have to solve for two vectors instead of only one.

Answer 1

See Transforming constraint of linear-fractional programming (Transformation to a linear program) for a recent closely related question.

1. Yes, apply Charnes-Cooper.
2. The same transformation applies whether you are maximizing or minimizing.
3. After the change of variables $y=t x$, you solve the LP for $y$ and take $x=y/t$ to recover the corresponding $x$.