Mixed-integer quadratic program

A mixed-integer quadratic program (MIQP) is an optimization problem of the form

.. math::

   \begin{array}{ll}
   \mbox{minimize}   & x^T Q x + q^T x + r \\
   \mbox{subject to} & x \in \mathcal{C}\\
   & x \in \mathbf{Z}^n,
   \end{array}

where :math:x \in \mathbf{Z}^n is the optimization variable (:math:\mathbf Z^n is the set of :math:n-dimensional vectors with integer-valued components), :math:Q \in \mathbf{S}_+^n (the set of :math:n \times n symmetric positive semidefinite matrices), :math:q \in \mathbf{R}^n, and :math:r \in \mathbf{R} are problem data, and :math:\mathcal C is some convex set.

An example of an MIQP is mixed-integer least squares, which has the form

.. math::

   \begin{array}{ll}
   \mbox{minimize}   & \|Ax-b\|_2^2 \\
   \mbox{subject to} & x \in \mathbf{Z}^n,
   \end{array}

where :math:x \in \mathbf{Z}^n is the optimization variable, and :math:A \in \mathbf{R}^{m \times n} and :math:b \in \mathbf{R}^{m} are the problem data. A solution :math:x^{\star} of this problem will be a vector in :math:\mathbf Z^n that minimizes :math:\|Ax-b\|_2^2.

Example

In the following code, we solve a mixed-integer least-squares problem with CVXPY. You need to install a mixed-integer nonlinear solver to run this example. CVXPY's preferred open-source mixed-integer nonlinear solver is SCIP. It can be installed with pip install pyscipopt or conda install -c conda-forge pyscipopt.

.. code:: python

import cvxpy as cp
import numpy as np

.. code:: python

# Generate a random problem
np.random.seed(0)
m, n= 40, 25

A = np.random.rand(m, n)
b = np.random.randn(m)

.. code:: python

# Construct a CVXPY problem
x = cp.Variable(n, integer=True)
objective = cp.Minimize(cp.sum_squares(A @ x - b))
prob = cp.Problem(objective)
prob.solve()

.. parsed-literal::

13.66000322824753

.. code:: python

print("Status: ", prob.status)
print("The optimal value is", prob.value)
print("A solution x is")
print(x.value)

.. parsed-literal::

Status:  optimal
The optimal value is 13.66000322824753
A solution x is
[-1.  1.  1. -1.  0.  0. -1. -2.  0.  0.  0.  1.  1.  0.  1.  0. -1. -1.
 -1.  0.  2. -1.  2.  0. -1.]