带约束的优化问题可被定义为：

在python中，可以使用scipy的optimize包进行求解，具体求解函数为linprog，下面举例说明求解方法：

假设问题被定义为：

首先，求解最大值问题，我们可以通过取负转换为求解最小值问题，包括不等式约束也是如此，那么该问题的python求解代码如下：

import numpy as np
from scipy.optimize import linprog
fun = np.array([-29.0, -45.0, 0.0, 0.0])
A_ub = np.array([[1.0, -1.0, -3.0, 0.0],
                [-2.0, 3.0, 7.0, -3.0]])
b_ub = np.array([5.0, -10.0])
A_eq = np.array([[2.0, 8.0, 1.0, 0.0],
                [4.0, 4.0, 0.0, 1.0]])
b_eq = np.array([60.0, 60.0])
x0_bounds = (0, None)
x1_bounds = (0, 5.0)
x2_bounds = (-np.inf, 0.5)  # +/- np.inf can be used instead of None
x3_bounds = (-3.0, None)
bounds = [x0_bounds, x1_bounds, x2_bounds, x3_bounds]
result = linprog(fun, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=bounds)
print(result.message)

上述代码会显示：

The problem is infeasible. (HiGHS Status 8: model_status is Infeasible; primal_status is At lower/fixed bound)

这说明问题无解，重新调整x1边界范围后，继续执行：

x1_bounds = (0, 6)
bounds = [x0_bounds, x1_bounds, x2_bounds, x3_bounds]
result = linprog(c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=bounds)
print(result.message)
print(result.x)
print(result.fun)

终端显示：

Optimization terminated successfully.
[ 9.41025641 5.17948718 -0.25641026 1.64102564]
-505.97435889013434

这说明问题求解成功，当$x_1=9.41025641,x_2=5.17948718,x_3=-0.25641026,x_4=1.64102564$ 目标函数取最大值，最大值为$505.97435889013434$.

注意将最大值优化问题转换为最小值优化问题时，结果要再取反！

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参考文献：

• https://docs.scipy.org/doc/scipy/tutorial/optimize.html#linear-programming-linprog