Recently, I have been trying to solve the following optimization problem:
$$ \begin{array}{cl} \text{maximize} & \frac{\left(c_1^T x\right)\left(c_2^T x\right)}{d^T x}\\ \text{subject to} & a^Tx\leq b \end{array} $$
where $a, c_1, c_2, d \in \Bbb R_{>0}^n$, $b > 0$, and $x_1, x_2, \dots, x_n \geq 0$.
The Lagrangian of the optimization problem can be written as $$ L=\frac{\left(c_1^T x\right)\left(c_2^T x\right)}{d^T x}+\theta\left(b-a^Tx\right)+\phi^Tx, $$ where $\theta\in \mathbb{R}$, $\theta\geq0$, $\phi$ is a $n\times1$ real vector and $\phi\geq0$. Then the KKT conditions are given by $$ \frac{c_1^Tx}{d^Tx}c_2+\frac{c_2^Tx}{d^Tx}c_1-\frac{c_1^Tx}{d^Tx}\frac{c_2^Tx}{d^Tx}d=a^T\theta-\phi, $$ $$ \theta\left(b-a^Tx\right)=0, $$ $$ \phi^Tx=0, $$ where $\theta\geq0$, $\phi\geq0$, $x\geq0$ and $a^Tx\leq b$.
Now I just want to do some relaxation to find an algorithm to obtain a sub-optimal solution. But I still have no clues. Could someone help me?
Thanks.
You can use de Lagrange Teorem and obtain a system with n-equations and n-variables.