Quick Start

Solving an optimization problem using Gurobi in Python usually involves the following steps:

1. Create an empty model

2. Add variables

3. Set the objective function

4. Add constraints

5. Optimize the model

6. Retrieve the solution

Create an empty model

gb.Model() is the most used class in Gurobi. It represents a mathematical optimization model.

import gurobipy as gb
from gurobipy import GRB

# Create an empty model
m = gb.Model("NewModel")

In the code above, m is an object of the class Model. The string "NewModel" is the name of the model.

Add variables

Class Model has a method called addVar() that adds a variable to the model. The method has the following signature:

# defuault arguments
x = m.addVar()

# arguments by name
y = m.addVar(lb=0, ub=1, obj=1.0, vtype=GRB.CONTINUOUS, name="y")

The arguments are:

• lb: lower bound of the variable

• ub: upper bound of the variable

• obj: coefficient of the variable in the objective function

• vtype: type of the variable. It can be GRB.CONTINUOUS, GRB.BINARY, or GRB.INTEGER

• name: name of the variable

By default, addVar(lb=0.0, ub=float('inf'), obj=0.0, vtype=GRB.CONTINUOUS, name="", column=None) is called.

In addition to the addVar() method, the Model class has the addVars() method and addMVar() method. The addVars() method adds multiple variables at once. The addMVar() method adds NumPy ndarray variables.

Set the objective function

The objective function is set using the setObjective() method. This method has two arguments: the expression of the objective function and the sense of the optimization. The sense can be GRB.MINIMIZE or GRB.MAXIMIZE.

# Set a objective function to maximize x + 2y
m.setObjective(x + 2 * y, GRB.MAXIMIZE)
# Set a objective function to minimize x^2 + y^2
m.setObjective(x**2 + y**2, GRB.MINIMIZE)

Note that setObjectiveN() can be used for multi-objective optimization.

Add constraints

Constraints are added using the addConstr() method. The method has two arguments: the expression of the constraint and the name of the constraint.

# Add a linear constraint x + y <= 5, named "c1"
m.addConstr(x + y <= 5, "c1")

# Add a quadratic constraint x^2 + y^2 <= 1, named "c2"
m.addConstr(x**2 + y**2 <= 1, "qc1")

# Add a ranged linear constraint x + y + z in [1, 2], named "rgc1"
m.addConstr(x + y + z == [1, 2], "rgc1")

# Add a matrix inequality constraint
m.addConstr(A @ x <= b, "matrix_inequality")

# Add a Absolute Value Constraint
m.addConstr(x == abs_(y), "abs1")

# Add a logical constraint
m.addConstr(z == and_(x, y, w), "logic1")

# Add a min or max constraint
m.addConstr(z == min_(x, y), "min1")

# Add an indicator constraint
m.addConstr((w == 1) >> (x + y <= 1), "ic1")

Optimize the model

Once we have added the variables, the objective function, and the constraints, we can optimize the model using the optimize() method.

m.optimize()

Retrieve the solution

After the optimization, we can retrieve the solution using the getVars() method.

for v in m.getVars():
    print(f"{v.VarName} {v.X:g}")

print(f"Obj: {m.ObjVal:g}")

Example: Solving a mixed-integer linear programming problem

Let's solve the following mixed-integer linear programming problem:

$$\begin{align*} \text{maximize} & \quad x + y + 2z \\ \text{subject to} & \quad x + 2y + 3z \leq 4 \\ & \quad x + y \geq 1 \\ & \quad x, y, z \in \{0, 1\} \end{align*}$$

import gurobipy as gb
from gurobipy import GRB

# Create an empty model
m = gb.Model("MILP")

# Add variables
x = m.addVar(vtype=GRB.BINARY, name="x")
y = m.addVar(vtype=GRB.BINARY, name="y")
z = m.addVar(vtype=GRB.BINARY, name="z")

# Set objective
m.setObjective(x + y + 2 * z, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x + 2 * y + 3 * z <= 4, "c1")
m.addConstr(x + y >= 1, "c2")

# Optimize
m.optimize()

# Print solution
for v in m.getVars():
    print(f"{v.VarName} {v.X:g}")

print(f"Obj: {m.ObjVal:g}")