使用Python求解带约束的最优化问题详解

题目：

1. 利用拉格朗日乘子法

#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 = 10 - x1*x1 - x2*x2 + alpha * (x1*x1 - x2) + beta * (x1 + x2)

#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_beta = diff(L, beta) #对乘子beta求导
dualCpt = alpha * (x1 * x1 - x2) #对偶互补条件

#求解KKT等式
aa = solve([difyL_x1, difyL_x2, difyL_beta, dualCpt], [x1, x2, alpha, beta])

#打印结果，还需验证alpha>=0和不等式约束<=0
for i in aa:
 if i[2] >= 0:
 if (i[0]**2 - i[1]) <= 0:
  print(i)

结果：

(-1, 1, 4, 6)
(0, 0, 0, 0)

2. scipy包里面的minimize函数求解

from scipy.optimize import minimize
import numpy as np 

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt 

#目标函数：
def func(args):
 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': 'ineq', 'fun': lambda x: x[1]-x[0]**2},
  {'type': 'eq', 'fun': lambda x: x[0]+x[1]})
 return cons 

#画三维模式图
def draw3D():
 fig = plt.figure()
 ax = Axes3D(fig)
 x_arange = np.arange(-5.0, 5.0)
 y_arange = np.arange(-5.0, 5.0)
 X, Y = np.meshgrid(x_arange, y_arange)
 Z1 = 10 - X**2 - Y**2
 Z2 = Y - X**2
 Z3 = X + Y
 plt.xlabel('x')
 plt.ylabel('y')
 ax.plot_surface(X, Y, Z1, rstride=1, cstride=1, cmap='rainbow')
 ax.plot_surface(X, Y, Z2, rstride=1, cstride=1, cmap='rainbow')
 ax.plot_surface(X, Y, Z3, rstride=1, cstride=1, cmap='rainbow')
 plt.show()

#画等高线图
def drawContour():
 x_arange = np.linspace(-3.0, 4.0, 256)
 y_arange = np.linspace(-3.0, 4.0, 256)
 X, Y = np.meshgrid(x_arange, y_arange)
 Z1 = 10 - X**2 - Y**2
 Z2 = Y - X**2
 Z3 = X + Y
 plt.xlabel('x')
 plt.ylabel('y')
 plt.contourf(X, Y, Z1, 8, alpha=0.75, cmap='rainbow')
 plt.contourf(X, Y, Z2, 8, alpha=0.75, cmap='rainbow')
 plt.contourf(X, Y, Z3, 8, alpha=0.75, cmap='rainbow')
 C1 = plt.contour(X, Y, Z1, 8, colors='black')
 C2 = plt.contour(X, Y, Z2, 8, colors='blue')
 C3 = plt.contour(X, Y, Z3, 8, colors='red')
 plt.clabel(C1, inline=1, fontsize=10)
 plt.clabel(C2, inline=1, fontsize=10)
 plt.clabel(C3, inline=1, fontsize=10)
 plt.show()

if __name__ == "__main__":
 args = ()
 args1 = ()
 cons = con(args1)
 x0 = np.array((1.0, 2.0)) #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.fun)
 print(res.success)
 print(res.x)

 # draw3D()
 drawContour()

结果：

7.99999990708696
True
[-1.00000002 1.00000002]

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