Given the generalized linear-fractional program:$$\text{Minimize}\;\; \max_{i}\Big|\frac{c_i^Tx+d_i}{e_i^Tx+f_i}\Big|$$$$\hspace{-5mm}\text{Subject to}\;\; e_i^Tx+f_i>0$$I convert this into the form:$$\text{Minimize}\;\; t$$$$\hspace{26mm}\text{Subject to}\;\; 0\leq c_i^Ty_i+d_iz_i\leq t$$$$\hspace{37mm}e_i^Ty+f_iz_i=1$$$$\hspace{24mm}z_i>0$$where $y_i=\frac{x}{e_i^Tx+f_i}$ and $z_i=\frac{1}{e_i^Tx+f_i}$.All this is for $i=1,...,k$.Yet when I put this into my solver (SCSSolver with the modeling tool Convex.jl), using actual data given to me in the problem, I get warnings thrown at me about the problem being likely degenerate and something about column pointers not being strictly increasing, none of which I understand.  I can probably go to a different forum to find out what these warnings mean, but is my mathematical approach to the problem correct?
Generalized linear-fractional program
optimization;convex optimization
To answer your question, yes, the mathematical approach you've taken is correct, and standard. You've combined two transformations:Transforming a linear fractional program to a linear program, as shown by Wikipedia; deducing the transformation is also a problem in some linear programming textbooks (for instance, Bertsimas and Tsitsiklis)Converting a $\max$ function objective into a linear program via constraints, which is typically covered explicitly in linear programming textbooksAbout the only thing I would change would be to change the $z_{i} > 0$ constraint to $z_{i} \geq 0$. Although the former is pedantically correct, no LP solver will actually implement such a constraint, so the closest realizable constraint is the latter. In the unlikely event that you obtain an optimal solution for which a $z_{i} = 0$, you could probably insert a cut that would exclude it.