A quadratic program is an optimization problem with a quadratic objective and affine equality and inequality constraints. A common standard form is the following: $$ \begin{array}{ll} \mbox{minimize} & (1/2)x^TPx + q^Tx\\ \mbox{subject to} & Gx \leq h \\ & Ax = b. \end{array} $$ Here $P \in \mathcal{S}^{n}_+$, $q \in \mathcal{R}^n$, $G \in \mathcal{R}^{m \times n}$, $h \in \mathcal{R}^m$, $A \in \mathcal{R}^{p \times n}$, and $b \in \mathcal{R}^p$ are problem data and $x \in \mathcal{R}^{n}$ is the optimization variable. The inequality constraint $Gx \leq h$ is elementwise.
A simple example of a quadratic program arises in finance. Suppose we have $n$ different stocks, an estimate $r \in \mathcal{R}^n$ of the expected return on each stock, and an estimate $\Sigma \in \mathcal{S}^{n}_+$ of the covariance of the returns. Then we solve the optimization problem $$ \begin{array}{ll} \mbox{minimize} & (1/2)x^T\Sigma x - r^Tx\\ \mbox{subject to} & x \geq 0 \\ & \mathbf{1}^Tx = 1, \end{array} $$ to find a nonnegative portfolio allocation $x \in \mathcal{R}^n_+$ that optimally balances expected return and variance of return.
When we solve a quadratic program, in addition to a solution $x^\star$, we obtain a dual solution $\lambda^\star$ corresponding to the inequality constraints. A positive entry $\lambda^\star_i$ indicates that the constraint $g_i^Tx \leq h_i$ holds with equality for $x^\star$ and suggests that changing $h_i$ would change the optimal value.
In the following code, we solve a quadratic program with CVXPY.
In [5]:
# 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)