# 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.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.        ]