How to convert the following if conditions to Linear integer programming constraints?

Question

These are the conditions:

if(x > 0)
{
    y >= a;
    z <= b;
}

It is quite easy to convert the conditions into Linear Programming constraints if x were binary variable. But I am not finding a way to do this.

Answer 1

You can do this in 2 steps

Step 1: Introduce a binary dummy variable

Since x is continuous, we can introduce a binary 0/1 dummy variable. Let's call it x_positive

if x>0 then we want x_positive =1. We can achieve that via the following constraint, where M is a very large number.

x < x_positive * M

Note that this forces x_positive to become 1, if x is itself positive. If x is negative, x_positive can be anything. (We can force it to be zero by adding it to the objective function with a tiny penalty of the appropriate sign.)

Step 2: Use the dummy variable to implement the next 2 constraints

In English: if x_positive = 1, then y >= a

However, if x_positive = 0, y can be anything (y > -inf)

y > a - M (1 - x_positive)

Similarly,

if x_positive = 1, then z <= b

z <= b + M * (1 - x_positive)

Both the linear constraints above will kick in if x>0 and will be trivially satisfied if x <=0.