共轭梯度法：

代码：

#导入模块
from sympy import *
import sympy as sp  #将导入的模块重新定义一个名字以便后续的程序进行使用
from numpy import *
import numpy as npdef main():#本例是利用共轭梯度法进行最优化x1,x2,alpha = symbols("x1,x2,alpha",real = True)f_fun = x1**2 + 25*x2**2x = np.mat(np.array([[2],[2]]))x0 = np.mat(np.array([[2],[2]]))f_diff_x01 = sp.diff(f_fun,x1).subs({x1:x0[0,0],x2:x0[1,0]})f_diff_x02 = sp.diff(f_fun,x2).subs({x1:x0[0,0],x2:x0[1,0]})f_diff_array = np.array([[f_diff_x01],[f_diff_x02]])f_diff_mat= np.mat(f_diff_array)d = -f_diff_matx_fun = x + alpha*df = (x_fun[0,0])**2 + 25*(x_fun[1,0])**2f_diff_alpha = sp.diff(f,alpha)alpha_solver = (solve([f_diff_alpha],[alpha]))[alpha]x_solver = x + alpha_solver*df_diff_x11 = float(sp.diff(f_fun,x1).subs({x1:x_solver[0,0],x2:x_solver[1,0]}))f_diff_x12 = float(sp.diff(f_fun,x2).subs({x1:x_solver[0,0],x2:x_solver[1,0]}))f_diff_array = np.array([[f_diff_x11],[f_diff_x12]])f_diff_mat= np.mat(f_diff_array)print("-------------------第一次--------------------")print("alpha:\n%s,\nx(1):\n%s,\nf_diff_1:\n%s\n"%(alpha_solver ,x_solver,f_diff_mat))beta = float(((f_diff_x11)**2 + (f_diff_x12)**2)/((f_diff_x01)**2 + (f_diff_x02)**2))d = (-f_diff_mat+beta*d)print(beta,d)x_fun = x_solver + alpha*df = (x_fun[0, 0]) ** 2 + 25 * (x_fun[1, 0]) ** 2f_diff_alpha = sp.diff(f,alpha)alpha_solver = (solve([f_diff_alpha],[alpha]))[alpha]x_solver = x + alpha_solver*df_diff_x11 = float(sp.diff(f_fun,x1).subs({x1:x_solver[0,0],x2:x_solver[1,0]}))f_diff_x12 = float(sp.diff(f_fun,x2).subs({x1:x_solver[0,0],x2:x_solver[1,0]}))f_diff_array = np.array([[f_diff_x11],[f_diff_x12]])f_diff_mat= np.mat(f_diff_array)print("-------------------第二次--------------------")print("alpha:\n%s,\nx(1):\n%s,\nf_diff_1:\n%s\n"%(alpha_solver ,x_solver,f_diff_mat))if __name__ == '__main__':main()

运行结果：

------------------------第1次迭代---------------------
alpha:
0.02003071803404582x:
[[1.91987712786382][-0.00307180340458224]]负梯度：
[[3.83975425572763][-0.153590170229112]]beta:
0.0014743712744378474d:
[[-3.84565174082538][0.00615304278532730]]判断条件：
14.76730268476948------------------------第2次迭代---------------------
alpha:
0.4992332268370619x:
[[-4.66293670342566e-15][3.21053947316408e-15]]负梯度：
[[-9.32587340685131e-15][1.60526973658204e-13]]beta:
0.0014743712744378474d:
[[ 9.32587341e-15][-1.60526974e-13]]判断条件：
2.585588118666227e-26进程已结束,退出代码0