Linear Programming - variable that equals the sign of an expression
divide-by-zero, linear-programming
Solution
You can't model `z = sign(x-c)` exactly with a linear program (because the constraints in an LP are restricted to linear combinations of variables).
However, you can model `sign` if you are willing to convert your linear program into a mixed integer program, you can model this with the following two constraints:
L*b <= x - c <= U*(1-b)
z = 1 - 2*b
Where `b` is a binary variable, and `L` and `U` are lower and upper bounds on the quantity `x-c`. If `b = 0`, we have `0 <= x - c <= U` and `z = 1`. If `b = 1`, we have `L <= x - c <= 0` and `z = 1 - 2*1 = -1`.
You can use a solver like Gurobi to solve mixed integer programs.
Problem
I am trying to write a linear program and need a variable z that equals the sign of x-c, where x is another variable, and c is a constant. I considered `z = (x-c)/|x-c|`. Unfortunately, if x=c, then this creates division by 0. I cannot use z=x-c, because I don't want to weight it by the magnitude of the difference between x and c. Does anyone know of a good way to express z so that it is the sign of x-c? Thank you for any help and suggestions!