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<=0for 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 minimizeimport numpy as np  from mpl_toolkits.mplot3d import Axes3Dfrom 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：

7.99999990708696True[-1.00000002 1.00000002]