How To Add Indicator Constraints In Pulp Python?

I have a problem that I don't know how to add indicator constraints in pulp. Can anyone help me? For example: I have a decision variable x[(i,j)], LpBinary and a continuous variabl

Solution 1:

Welcome to SO!

My interpretation of your question is that you have binary variables x[(i,j)], and continuos variables u[i]. When x[(i,j)]==1 then you want to enforce a constraint as follows u[i] + q[j] == u[j]. If x[(i,j)]==0 then no such constraint is enforced.

This can be done as follows:

for i in set_I:
    for j in set_J:
        u[j] >= u[i] + q[j] - (1 - x[(i,j)])*M
        u[j] <= u[i] + q[j] + (1 - x[(i,j)])*M

Where M is a value that is a bit bigger than the largest possible range in u[i] values + the largest possible q[j] value. To understand why this works consider the two cases, first if x[(i,j)]==1 these constraints become:

    u[j] >= u[i] + q[j]
    u[j] <= u[i] + q[j]

Which can be abbreviated as: u[j] == u[i] + q[j], the constraint you want in the x[(i,j)]==1 case.

In the x[(i,j)]==0 case, these constraints become:

    u[j] >= u[i] + q[j] - M
    u[j] <= u[i] + q[j] + M

Recall that M is a large number, what we are saying is u[j] >= some_value - large_number, which provided you choose M so that its large enough will not have any effect at all (as required). Similarly the constraint u[j] <= some_value + large_number has not effect provided M is sufficiently large.