CVXPY in Python

CVXPY in Python

Here’s a basic example of using $CVXPY$, a $Python$ library for convex optimization.

This example solves a simple linear programming problem:

Problem

Minimize the function $( c^T x )$ subject to the constraint $( Ax \leq b )$ and $( x \geq 0 )$, where:

  • $( c )$ is a vector of coefficients for the objective function.
  • $( A )$ is a matrix of coefficients for the inequality constraints.
  • $( b )$ is a vector representing the upper bounds for the constraints.

Sample Code

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import cvxpy as cp
import numpy as np

# Define the data for the problem
c = np.array([1, 2])        # Coefficients for the objective function
A = np.array([[1, 1], [1, -1], [-1, 2]])  # Coefficients for the constraints
b = np.array([2, 1, 2])     # Right-hand side values for the constraints

# Define the optimization variables
x = cp.Variable(2)

# Define the objective function: minimize c^T x
objective = cp.Minimize(c @ x)

# Define the constraints: Ax <= b and x >= 0
constraints = [A @ x <= b, x >= 0]

# Formulate the problem
problem = cp.Problem(objective, constraints)

# Solve the problem
problem.solve()

# Print the results
print("Status:", problem.status)
print("Optimal value of the objective function:", problem.value)
print("Optimal values of the variables x:", x.value)

Explanation

  • Objective Function:
    c @ x is the dot product of the vector c with the variable vector x. We aim to minimize this value.
  • Constraints:
    A @ x <= b represents the inequality constraints, and x >= 0 ensures that the variables are non-negative.
  • Optimization:
    problem.solve() solves the optimization problem, and the optimal solution is stored in x.value.

Output

When you run this code, it will output the status of the optimization (e.g., “optimal”), the optimal value of the objective function, and the optimal values of the decision variables $( x )$.

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Status: optimal
Optimal value of the objective function: 3.698490338406144e-10
Optimal values of the variables x: [3.59660015e-10 5.09450927e-12]

This example is a basic introduction, but $CVXPY$ can handle more complex problems, including quadratic programming, mixed-integer programming, and other types of convex optimization problems.