How do I model logical and general expressions?

General Constraints can be used to model many types of logical expressions, like AND and OR relationships. More complex logical expressions can be created via combinations of general constraints.

For example, to model an AND expression in Python, you can use the Model.addGenConstrAnd() method:

# x5 = and(x1, x3, x4)
model.addGenConstrAnd(x5, [x1, x3, x4], "and_constr")

or, using the overloaded form via the and_() helper function:

model.addConstr(x5 == and_([x1, x3, x4]), "and_constr")

To work with general expressions within constraints, you may sometimes need to add intermediate decision variables. For example, the abs_() function is used to set one variable equal to the absolute value of another. To model the constraint \( v = | x - y | \), define a free auxiliary variable \( u \) and set it equal to \( x − y \). Then, set \( v \) equal to \( | u | \). For example, in Python:

import gurobipy as gp

model = gp.Model()

x = model.addVar(name="x")
y = model.addVar(name="y")
v = model.addVar(name="v")

# v = |x - y|
u = model.addVar(lb=-gp.GRB.INFINITY, name="u")
model.addConstr(u == x - y)
model.addConstr(v == gp.abs_(u))

Further information

• 3D Tic-Tac-Toe Jupyter notebook
• What types of models can Gurobi solve?
• How do I model piecewise-linear functions?