I am a newbie in the domain of optimization. I am trying to solve an optimization problem, which has a set of constraints. Since the number of constraints follows similar pattern I intend to reduce the number of constraints.
My constraints look as below.
$w_1>c_1*w_2$
$w_1>c_2*w_2$
$w_1>c_3*w_2$
$w_1>c_4*w_2$
$w_1>c_5*w_2$
$w_1>c_6*w_2$
$w_1>c_7*w_2$
Here, $c_{i}$ is a constant term and $w_{i}$ is a variable.
This set of constraints have been reduced to two constraints. New set of constraints are
$w_1 > c_{max}*w_2$
$w_1 > c_{min}*w_2$
Here, I assumed the possibilities of positive and negative values of $w_2$ to form above-mentioned constraints. Am I doing it, right?
Secondly,
I am also working with a larger problem, and will have constraints similar to,
$w_1 > c_1*w_2 + c_{11}*w_3$
(having 10000 such constraints)
I tried to follow the same strategy to reduce these set of constraints, and hence got
$w_1 > c_{max}*w_2 + c_{max}*w_3$
$w_1 > c_{max}*w_2 + c_{min}*w_3$
$w_1 > c_{min}*w_2 + c_{max}*w_3$
$w_1 > c_{min}*w_2 + c_{min}*w_3$
Again considering the positive and negative values of $w_{i}$. The above mentioned constraints are not the perfect formulation and its not ideal too as it is evident that the set of constraints increases with n following the relation: $2^{n-1}$, where n is the number of variables. So, for n=70 variables and having 100 such constraints this idea will lead to $2^{70}$ reduced constraints, whereas originally it was only 100.
I am looking an alternative way to reduce them, any suggestion would be highly appreciable.