If it is possible to linearize no-else if statement as MIP constraint?

Question

Assume that I have the following constraint:

IF d == 1 THEN x >= 1;

d : binary

x : integer

but there is no else, means that IF d == 0, there will be no constraint for x.

Is such constraint convertable to MIP linear constraint?

Answer 1

This can be generally modelled in mixed integer programming with a "Big M Constraint":

from pulp import *

d = LpVariable("d", cat='Binary')
x = LpVariable("x", lowBound=0, cat='Integer')

M = 100000

constraint = x >= 1 - M * (1 - d)

If d==1 it turns into x>=1, else it's x>=1-M where M is a suitably chosen large number.