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.]