python 实现拉格朗日乘子法

from scipy.optimize import minimize
import numpy as np 
 
#目标函数：
def func(args):
    fun = lambda x: 60 - 10*x[0] - 4*x[1] + x[0]**2 + x[1]**2 - x[0]*x[1]
    #fun = lambda x: 10 - x[0]**2 - x[1]**2
    return fun
 
#约束条件，包括等式约束和不等式约束
def con(args):
    cons = ({'type': 'eq', 'fun': lambda x: x[0]+x[1]-8})
    #cons = ({'type': 'ineq', 'fun': lambda x: x[1]-x[0]**2},
    #        {'type': 'eq', 'fun': lambda x: x[0]+x[1]})
    return cons 
 
if __name__ == "__main__":
    args = ()
    args1 = ()
    cons = con(args1)
    x0 = np.array((2.0, 1.0))  #设置初始值，初始值的设置很重要，很容易收敛到另外的极值点中，建议多试几个值
    
    #求解#
    res = minimize(func(args), x0, method='SLSQP', constraints=cons)
    print(res.success)
    print("x1=",res.x[0],";  x2=",res.x[1])
    print("最优解为：",res.fun)

x1= 4.999999943481969 ;  x2= 3.000000056518032
最优解为： 17.000000000000007

#导入sympy包，用于求导，方程组求解等等
from sympy import * 
 
#设置变量
x1 = symbols("x1")
x2 = symbols("x2")
alpha = symbols("alpha")
#beta = symbols("beta")
 
#构造拉格朗日等式
L = 60 - 10*x1 - 4*x2 + x1*x1 + x2*x2 - x1*x2 - alpha * (x1 + x2 - 8)
 
#求导，构造KKT条件
difyL_x1 = diff(L, x1)  #对变量x1求导
difyL_x2 = diff(L, x2)  #对变量x2求导
difyL_alpha = diff(L, alpha) #对alpha求导
 
#求解KKT等式
aa = solve([difyL_x1, difyL_x2, difyL_alpha], [x1, x2, alpha])
print(aa)
x1=aa.get(x1)
x2=aa.get(x2)
alpha=aa.get(alpha)
print("最优解为：",60 - 10*x1 - 4*x2 + x1*x1 + x2*x2 - x1*x2 - alpha * (x1 + x2 - 8))

{alpha: -3, x1: 5, x2: 3}
最优解为： 17