Quadratic program

A quadratic program is an optimization problem with a quadratic objective and affine equality and inequality constraints. A common standard form is the following:

.. math::

   \begin{array}{ll}
   \mbox{minimize}   & (1/2)x^TPx + q^Tx\\
   \mbox{subject to} & Gx \leq h \\
                     & Ax = b.
   \end{array}

Here :math:P \in \mathcal{S}^{n}_+, :math:q \in \mathcal{R}^n, :math:G \in \mathcal{R}^{m \times n}, :math:h \in \mathcal{R}^m, :math:A \in \mathcal{R}^{p \times n}, and :math:b \in \mathcal{R}^p are problem data and :math:x \in \mathcal{R}^{n} is the optimization variable. The inequality constraint :math:Gx \leq h is elementwise.

A simple example of a quadratic program arises in finance. Suppose we have :math:n different stocks, an estimate :math:r \in \mathcal{R}^n of the expected return on each stock, and an estimate :math:\Sigma \in \mathcal{S}^{n}_+ of the covariance of the returns. Then we solve the optimization problem

.. math::

   \begin{array}{ll}
   \mbox{minimize}   & (1/2)x^T\Sigma x - r^Tx\\
   \mbox{subject to} & x \geq 0 \\
                     & \mathbf{1}^Tx = 1,
   \end{array}

to find a portfolio allocation :math:x \in \mathcal{R}^n_+ that optimally balances expected return and variance of return.

When we solve a quadratic program, in addition to a solution :math:x^\star, we obtain a dual solution :math:\lambda^\star corresponding to the inequality constraints. A positive entry :math:\lambda^\star_i indicates that the constraint :math:g_i^Tx \leq h_i holds with equality for :math:x^\star and suggests that changing :math:h_i would change the optimal value.

Example

In the following code, we solve a quadratic program with CVXPY.

.. code:: python

# Import packages.
import cvxpy as cp
import numpy as np

# Generate a random non-trivial quadratic program.
m = 15
n = 10
p = 5
np.random.seed(1)
P = np.random.randn(n, n)
P = P.T @ P
q = np.random.randn(n)
G = np.random.randn(m, n)
h = G @ np.random.randn(n)
A = np.random.randn(p, n)
b = np.random.randn(p)

# Define and solve the CVXPY problem.
x = cp.Variable(n)
prob = cp.Problem(cp.Minimize((1/2)*cp.quad_form(x, P) + q.T @ x),
                 [G @ x <= h,
                  A @ x == b])
prob.solve()

# Print result.
print("\nThe optimal value is", prob.value)
print("A solution x is")
print(x.value)
print("A dual solution corresponding to the inequality constraints is")
print(prob.constraints[0].dual_value)

.. parsed-literal::

The optimal value is 86.89141585569918
A solution x is
[-1.68244521  0.29769913 -2.38772183 -2.79986015  1.18270433 -0.20911897
 -4.50993526  3.76683701 -0.45770675 -3.78589638]
A dual solution corresponding to the inequality constraints is
[ 0.          0.          0.          0.          0.         10.45538054
  0.          0.          0.         39.67365045  0.          0.
  0.         20.79927156  6.54115873]