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：

三种最优化方法的迭代次数对比：

以上就是共轭梯度法在Python中的实现，是不是很有意思呢。