Linear Programming - variable that equals the sign of an expression

Question

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!

Answer 1

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.

Answer 2

For k » 1 this is a smooth approximation of the sign function:

Also

when ε → 0

These two approximations haven't the division by 0 issue but now you must tune a parameter.

---

In some languages (e.g. C++ / C) you can simply write something like this:

double sgn(double x)
{
  return (x > 0.0) - (x < 0.0);
}

Anyway consider that many environments / languages already have a sign function, e.g.

• Sign[x] in Mathematica
• sign(x) in Matlab
• Math.signum(x) in Java
• sign(1, x) in Fortran
• sign(x) in R

Pay close attention to what happens when x is equal to 0 (e.g. the Fortran function will return 1, with other languages you'll get 0).