I need help in formulating an optimization problem. I have a system of equations as follows:
$c_1x_1+c_2x_2+c_3x_3=1$
$b_1x_1+b_2x_2+b_3x_3=1$
$a_1x_1+a_2x_2+a_3x_3=1$
In my case the system is not necessarily in three variables (the number of variables can increase).
The chosen $x_1$, $x_2$, $x_3$ should also satisfy the following system of equations:
$c_1'x_1+c_2'x_2+c_3'x_3<1$
$b_1'x_1+b_2'x_2+b_3'x_3<1$
$a_1'x_1+a_2'x_2+a_3'x_3<1$
where
$0<c_1'<c_1$, $0<c_2'<c_2$, $0<c_3'<c_3$
$0<b_1'<b_1$, $0<b_2'<b_2$, $0<b_3'<b_3$
$0<a_1'<a_1$, $0<a_2'<a_2$, $0<a_3'<a_3$
I have no idea where should I look for a solution. Any pointers in the right direction would also be appreciated.
UPDATE: I understand that the constraints $0<c_1'<c_1$, $0<c_2'<c_2$, $0<c_3'<c_3$ would form a rectangular area and the value of $x_1$, $x_2$, $x_3$ should be chosen so that over this rectangular area the inequalities should be met. But I don't know how to formulate the constraints into an equation.