ILP - “变量是否等于另一个变量”

Question

In an ILP, given two variables x and y, is it possible to define a variable z where

z = (x==y)?

Meaning, if x equals y then z = 1. Else, z = 0.

x and y are integer variables bounded between (1, M).

Answer 1

If you know that x <= y , then you can use z in {0,1}, x+Mz >= y, x+z <= y .

If you don't know which of x and y is minimum, you can do it with more work, by adding a variable minxy which takes the value of the minimum. You need to introduce another new variable (which I call a ) to do this.

Introduce a variable a that's 0 if x<=y, 1 if y<=x (it could be either 0 or 1 if x==y):

a in {0,1}, x-y <= Ma, y-x <= M(1-a)

Introduce a variable minxy that's the minimum of x and y :

minxy <= x
minxy <= y
minxy >= x - Ma
minxy >= y - M(1-a)

Then you can define your z :

minxy + Mz >= x + y - minxy
minxy + z <= x + y - minxy

(Noting that max(x,y) is x + y - minxy ).