How to formulate multiple conditional logical (AND and OR) constraints

Answered

Babooshka Shavazipour

September 15, 2023 10:05

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I am wondering what would be the efficient way of formulating the following constraint in (Python) Gurobi:

# The big-M
M = 10**9
# Constraints
for i in range(n):
    for k inrange(n):
        if (i!=k) and (x[i]==1) and (x[k]==1):
            # OR constraints already linearized via Big-M
            px[i] - px[k] +M*eta <= M-l[i]
            px[k] - px[i] -M*eta <= -l[k]

x_i  (i=1,...,n), eta, and gamma are binary variables and px_i (i=1,...,n) are integer variables. 

Related to

• General Constraints

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• 

  Maliheh Aramon

  • Gurobi Staff

  September 16, 2023 00:43

  Hi, 

  You can define an auxiliary variable \(z_{ik} = x_i \land x_k\, ~~ \forall i, k\). The logical And constraints can be implemented using the Model.addGenConstrAnd() method in Gurobi Python API. 

  You can then use the Model.addGenConstrIndicator() method to implement the following two constraints for all \(i\) and \(k\) indices where \(i \neq k\):

  • \(\mbox{if}~ z_{ik} = 1 \rightarrow px_i - px_k + M \eta \leq M - l_i\)
  • \(\mbox{if}~ z_{ik} = 1 \rightarrow px_i - px_k - M \eta \leq - l_k\)

  Best regards,

  Maliheh

  0

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