python实现共轭梯度法

Rose ·

更新时间:2024-09-21

· 946 次阅读

共轭梯度法是介于最速下降法与牛顿法之间的一个方法，它仅需利用一阶导数信息，但克服了最速下降法收敛慢的缺点，又避免了牛顿法需要存储和计算Hesse矩阵并求逆的缺点，共轭梯度法不仅是解决大型线性方程组最有用的方法之一，也是解大型非线性最优化最有效的算法之一。在各种优化算法中，共轭梯度法是非常重要的一种。其优点是所需存储量小，具有步收敛性，稳定性高，而且不需要任何外来参数。

算法步骤：


import random
import numpy as np
import matplotlib.pyplot as plt
def goldsteinsearch(f,df,d,x,alpham,rho,t):
 '''
 线性搜索子函数
 数f，导数df，当前迭代点x和当前搜索方向d，t试探系数>1，
 '''
 flag = 0
 a = 0
 b = alpham
 fk = f(x)
 gk = df(x)
 phi0 = fk
 dphi0 = np.dot(gk, d)
 alpha=b*random.uniform(0,1)
 while(flag==0):
  newfk = f(x + alpha * d)
  phi = newfk
  # print(phi,phi0,rho,alpha ,dphi0)
  if (phi - phi0 )<= (rho * alpha * dphi0):
   if (phi - phi0) >= ((1 - rho) * alpha * dphi0):
    flag = 1
   else:
    a = alpha
    b = b
    if (b < alpham):
     alpha = (a + b) / 2
    else:
     alpha = t * alpha
  else:
   a = a
   b = alpha
   alpha = (a + b) / 2
 return alpha
def Wolfesearch(f,df,d,x,alpham,rho,t):
 '''
 线性搜索子函数
 数f，导数df，当前迭代点x和当前搜索方向d
 σ∈(ρ,1)=0.75
 '''
 sigma=0.75
 flag = 0
 a = 0
 b = alpham
 fk = f(x)
 gk = df(x)
 phi0 = fk
 dphi0 = np.dot(gk, d)
 alpha=b*random.uniform(0,1)
 while(flag==0):
  newfk = f(x + alpha * d)
  phi = newfk
  # print(phi,phi0,rho,alpha ,dphi0)
  if (phi - phi0 )<= (rho * alpha * dphi0):
   # if abs(np.dot(df(x + alpha * d),d))<=-sigma*dphi0:
   if (phi - phi0) >= ((1 - rho) * alpha * dphi0):
    flag = 1
   else:
    a = alpha
    b = b
    if (b < alpham):
     alpha = (a + b) / 2
    else:
     alpha = t * alpha
  else:
   a = a
   b = alpha
   alpha = (a + b) / 2
 return alpha
def frcg(fun,gfun,x0):
 # x0是初始点，fun和gfun分别是目标函数和梯度
 # x,val分别是近似最优点和最优值，k是迭代次数
 # dk是搜索方向，gk是梯度方向
 # epsilon是预设精度，np.linalg.norm(gk)求取向量的二范数
 maxk = 5000
 rho = 0.6
 sigma = 0.4
 k = 0
 epsilon = 1e-5
 n = np.shape(x0)[0]
 itern = 0
 W = np.zeros((2, 20000))
 f = open("共轭.txt", 'w')
 while k < maxk:
   W[:, k] = x0
   gk = gfun(x0)
   itern += 1
   itern %= n
   if itern == 1:
    dk = -gk
   else:
    beta = 1.0 * np.dot(gk, gk) / np.dot(g0, g0)
    dk = -gk + beta * d0
    gd = np.dot(gk, dk)
    if gd >= 0.0:
     dk = -gk
   if np.linalg.norm(gk) < epsilon:
    break
   alpha=goldsteinsearch(fun,gfun,dk,x0,1,0.1,2)
   # alpha=Wolfesearch(fun,gfun,dk,x0,1,0.1,2)
   x0+=alpha*dk
   f.write(str(k)+' '+str(np.linalg.norm(gk))+"\n")
   print(k,alpha)
   g0 = gk
   d0 = dk
   k += 1
 W = W[:, 0:k+1] # 记录迭代点
 return [x0, fun(x0), k,W]
def fun(x):
 return 100 * (x[1] - x[0] ** 2) ** 2 + (1 - x[0]) ** 2
def gfun(x):
 return np.array([-400 * x[0] * (x[1] - x[0] ** 2) - 2 * (1 - x[0]), 200 * (x[1] - x[0] ** 2)])
if __name__=="__main__":
 X1 = np.arange(-1.5, 1.5 + 0.05, 0.05)
 X2 = np.arange(-3.5, 4 + 0.05, 0.05)
 [x1, x2] = np.meshgrid(X1, X2)
 f = 100 * (x2 - x1 ** 2) ** 2 + (1 - x1) ** 2 # 给定的函数
 plt.contour(x1, x2, f, 20) # 画出函数的20条轮廓线
 x0 = np.array([-1.2, 1])
 x=frcg(fun,gfun,x0)
 print(x[0],x[2])
 # [1.00318532 1.00639618]
 W=x[3]
 # print(W[:, :])
 plt.plot(W[0, :], W[1, :], 'g*-') # 画出迭代点收敛的轨迹
 plt.show()

代码中求最优步长用得是goldsteinsearch方法，另外的Wolfesearch是试验的部分，在本段程序中不起作用。

迭代轨迹：