How to maximize $$Y = \sum_{i=0}^n \frac{a_ix_i}{b_ix_i+c_i}$$ Given the following constraints:
- $\sum_{i=0}^n x_i = X$ (X is a constant)
- $x_0, x_1, ..., x_n >= 0$
In the above, $a, b, c$ are vectors with known input (all integers). $X$ is also known. Thus, I need to find how to split $X$ to optimize $Y$.
I have tried using the Linear-Fractional programming to transform it into a system of linear equations, over which to use the Simplex Method, but I have a hard time translating the constraints. And I am not even sure that is the correct approach to this problem.
Because of the constraint across $i$, the Charnes-Cooper transformation from linear fractional programming to linear programming does not apply here, but you can instead transform to second-order cone programming, as shown in my answer here: https://or.stackexchange.com/a/8959/500