I have an linear-fractional program defined like this:
\begin{equation} \begin{aligned} \max_{x} \quad & \frac{c^{T}x + \alpha}{d^{T}x + \beta}\\ \textrm{s.t.} \quad & Ax \geq b\\ \end{aligned} \end{equation}
However, as you can see my optimization equation, constraint is slightly different from https://en.wikipedia.org/wiki/Linear-fractional_programming
Mine uses greater than operator in constraint, but wikipedia's equation uses less than operator in constraint.
So I have a question about transforming constraints in linear fractional programming.
In wikipedia page section (Transformation to a linear program), it transforms original formula like below. (This equation is same as widipedia)
\begin{equation} \begin{aligned} \max_{x} \quad & c^{T}y + \alpha t\\ \textrm{s.t.} \quad & Ay \leq bt\\ &d^{T}y + \beta t = 1 \\ &t \geq 0 \end{aligned} \end{equation}
But my problem's constraint is slightly different, so I changed the first constraint ( $Ay \leq bt$ ) to ( $Ay \geq bt$ ).
Is it okay to change the first constraint ( $Ay \leq bt$ ) to ( $Ay \geq bt$ ) ?
I only changed the lessthan operator to greaterthan operator.
Or how can I change my linear-fractional program to linear program?
PS: sorry for my english.. I'm not familiar with english.
Thank you very much!
Yes, what you have done is correct. Both $\le$ and $\ge$ are acceptable for linear programs.