CVXPY
🔬 线性规划
We consider thefollowing problem
\[ \begin{align*}
& \underset{x}{\min}~-3x_1-2x_2 \\
& {\rm s.t.}~3x_1 + 4x_2 \le 7 \\
& \quad~~2x_1 + x_2 \le 3 \\
& \quad~~-3x_1 + 2x_2 = 2 \\
& \quad~~0 \le x_1,x_2 \le 10 \\
\end{align*}
\]
import numpy as np
import cvxpy as cp
import time
# Seed for reproducibility
np.random.seed(1)
# Define parameters
c = np.array([-3, -2])
A_ub = np.array([[3, 4], [2, 1]])
b_ub = np.array([7, 3])
# Equality constraint
Aeq = np.array([[-3, 2]])
beq = np.array([2])
# Bounds for variables
lb = np.array([0, 0])
ub = np.array([10, 10])
# Define the problem
n = 2
x = cp.Variable(n)
prob = cp.Problem(cp.Minimize(c @ x),
[A_ub @ x <= b_ub,
Aeq @ x == beq,
x >= lb,
x <= ub])
# Solve the problem
result = prob.solve()
# Print results
print("Optimal value:", result)
print("Optimal point:", x.value)
Optimal value: -4.0000000001132126
Optimal point: [0.33333333 1.5 ]
🔬 二次规划
We consider the following problem
\[ \min f(x) = \frac{1}{2}x_1^2 + x_2^2 -x_1x_2 -2x_1 -6x_2
= \frac{1}{2}x^{T}Hx + c^Tx
\]
The constraint condition is
\[ \begin{cases}
x_1 + x_2 \le 2\\
-x_1 + 2x_2 \le 2 \\
2x_1 + x_2 \le 3 \\
x_1 \ge 0,x_2 \ge 0.
\end{cases}
\]
Where
\[ H =
\begin{bmatrix}
1 & -1 \\
-1 & 2
\end{bmatrix},
c =
\begin{bmatrix}
-2 \\
-6
\end{bmatrix},
x =
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
\]
H = np.array([[1, -1], [-1, 2]])
c = np.array([-2, -6])
A = np.array([[1, 1], [-1, 2], [2, 1]])
b = np.array([2, 2, 3])
lb = np.array([0, 0])
start_time = time.time()
# Define and solve the CVXPY problem
x = cp.Variable(2)
prob = cp.Problem(cp.Minimize((1/2) * cp.quad_form(x, H) + c @ x),
[A @ x <= b,
x >= lb])
result = prob.solve()
end_time = time.time()
# Print the results
print("Optimal value:", result)
print("Optimal point:", x.value)
print("Time:",end_time-start_time)
Optimal value: -8.222222222222223
Optimal point: [0.66666667 1.33333333]
Time: 0.007505178451538086
