#導(dǎo)入sympy包，用于求導(dǎo)，方程組求解等等
from sympy import * 
 
#設(shè)置變量
x1 = symbols("x1")
x2 = symbols("x2")
alpha = symbols("alpha")
beta = symbols("beta")
 
#構(gòu)造拉格朗日等式
L = 10 - x1*x1 - x2*x2 + alpha * (x1*x1 - x2) + beta * (x1 + x2)
 
#求導(dǎo)，構(gòu)造KKT條件
difyL_x1 = diff(L, x1) #對(duì)變量x1求導(dǎo)
difyL_x2 = diff(L, x2) #對(duì)變量x2求導(dǎo)
difyL_beta = diff(L, beta) #對(duì)乘子beta求導(dǎo)
dualCpt = alpha * (x1 * x1 - x2) #對(duì)偶互補(bǔ)條件
 
#求解KKT等式
aa = solve([difyL_x1, difyL_x2, difyL_beta, dualCpt], [x1, x2, alpha, beta])
 
#打印結(jié)果，還需驗(yàn)證alpha>=0和不等式約束<=0
for i in aa:
 if i[2] >= 0:
 if (i[0]**2 - i[1]) <= 0:
  print(i)

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

from scipy.optimize import minimize
import numpy as np 
 
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt 
 
#目標(biāo)函數(shù)：
def func(args):
 fun = lambda x: 10 - x[0]**2 - x[1]**2
 return fun
 
#約束條件，包括等式約束和不等式約束
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 
 
#畫(huà)三維模式圖
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()
 
#畫(huà)等高線圖
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)) #設(shè)置初始值，初始值的設(shè)置很重要，很容易收斂到另外的極值點(diǎn)中，建議多試幾個(gè)值
 
 #求解#
 res = minimize(func(args), x0, method='SLSQP', constraints=cons)
 #####
 print(res.fun)
 print(res.success)
 print(res.x)
 
 # draw3D()
 drawContour()

7.99999990708696
True
[-1.00000002 1.00000002]