Linear program

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

\[\begin{split}\begin{array}{ll} \mbox{minimize} & c^Tx \\ \mbox{subject to} & Ax \leq b. \end{array}\end{split}\]

Here \(A \in \mathcal{R}^{m \times n}\), \(b \in \mathcal{R}^m\), and \(c \in \mathcal{R}^n\) are problem data and \(x \in \mathcal{R}^{n}\) is the optimization variable. The inequality constraint \(Ax \leq b\) is elementwise.

For example, we might have \(n\) different products, each constructed out of \(m\) components. Each entry \(A_{ij}\) is the amount of component \(i\) required to build one unit of product \(j\). Each entry \(b_i\) is the total amount of component \(i\) available. We lose \(c_j\) for each unit of product \(j\) (\(c_j < 0\) indicates profit). Our goal then is to choose how many units of each product \(j\) to make, \(x_j\), in order to minimize loss without exceeding our budget for any component.

In addition to a solution \(x^\star\), we obtain a dual solution \(\lambda^\star\). A positive entry \(\lambda^\star_i\) indicates that the constraint \(a_i^Tx \leq b_i\) holds with equality for \(x^\star\) and suggests that changing \(b_i\) would change the optimal value.

Example

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

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

# Generate a random non-trivial linear program.
m = 15
n = 10
np.random.seed(1)
s0 = np.random.randn(m)
lamb0 = np.maximum(-s0, 0)
s0 = np.maximum(s0, 0)
x0 = np.random.randn(n)
A = np.random.randn(m, n)
b = A @ x0 + s0
c = -A.T @ lamb0

# Define and solve the CVXPY problem.
x = cp.Variable(n)
prob = cp.Problem(cp.Minimize(c.T@x),
                 [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 is")
print(prob.constraints[0].dual_value)

The optimal value is -15.220912604467838
A solution x is
[-1.10131657 -0.16370661 -0.89711643  0.03228613  0.60662428 -1.12655967
  1.12985839  0.88200333  0.49089264  0.89851057]
A dual solution is
[0.         0.61175641 0.52817175 1.07296862 0.         2.3015387
 0.         0.7612069  0.         0.24937038 0.         2.06014071
 0.3224172  0.38405435 0.        ]