原理什么的就不說了，只有代碼。。。

十元二次目標函數，求極小值點。

1.最速下降法

import random

import numpy as np

import matplotlib.pyplot as plt

from sympy import *

def goldsteinsearch(f, df, d, x, alpham, rho, t):

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 steepest(fun, gfun,x0):

c = 0.00001

imax = 100

i = 1

x = x0

grad = gfun(x)

delta = sum(grad**2)

while i < imax and delta > c:

p = -gfun(x)

alpha = goldsteinsearch(fun, gfun, p, x, 1, 0.1, 2)

x = x + alpha * p

print(i, np.array((i, delta)))

grad = gfun(x)

delta = sum(grad**2)

i = i + 1

return x, fun(x), i, delta

#目標函數

fun = lambda x: (x[0]-1)**2+(x[1]-1)**2+(x[2]-2)**2+(x[3]-3)**2+(x[4]-4)**2+(x[5]-5)**2+(x[6]-6)**2+(x[7]-7)**2+(x[8]-8)**2+(x[9]-9)**2+x[0]*x[1]+x[2]*x[3]+x[4]*x[5]+x[6]*x[7]+x[8]*x[9]

#一階導數

gfun = lambda x: np.array([2*x[0] + x[1] - 2, x[0] + 2*x[1] - 2, 2*x[2] + x[3] - 4, x[2] + 2*x[3] - 6, 2*x[4] + x[5] - 8, x[4] + 2*x[5] - 10, 2*x[6] + x[7] - 12, x[6] + 2*x[7] - 14, 2*x[8] + x[9] - 16, x[8] + 2*x[9] - 18])

if __name__ == "__main__":

x0 = np.array([1.0,1.0,1.0,3.0,1.0,3.0,1.0,1.0,2.0,3.0])

steepest(fun, gfun,x0)

output = steepest(fun, gfun,x0)

print ("迭代次數:" output [2])

print ("近似最優解:" output [0])

print ("函數極小值點:" output [1])

print ("最終迭代誤差:" output [3])

2.牛頓法

import random

import numpy as np

import matplotlib.pyplot as plt

def dampnm(fun, gfun, hess, x0):

maxk = 500

rho = 0.55

sigma = 0.4

k = 0

delta = 1e-5

_delta=delta

while k < maxk:

gk = gfun(x0)

Gk = hess(x0)

dk = -1.0 * np.linalg.solve(Gk, gk)

print(k, np.linalg.norm(dk))

if np.linalg.norm(dk) < delta:

_delta = np.linalg.norm(dk)

break

x0 += dk

k += 1

return x0, fun(x0), k,_delta

#目標函數

fun = lambda x: (x[0]-1)**2+(x[1]-1)**2+(x[2]-2)**2+(x[3]-3)**2+(x[4]-4)**2+(x[5]-5)**2+(x[6]-6)**2+(x[7]-7)**2+(x[8]-8)**2+(x[9]-9)**2+x[0]*x[1]+x[2]*x[3]+x[4]*x[5]+x[6]*x[7]+x[8]*x[9]

#一階導數

gfun = lambda x: np.array([2*x[0] + x[1] - 2, x[0] + 2*x[1] - 2, 2*x[2] + x[3] - 4, x[2] + 2*x[3] - 6, 2*x[4] + x[5] - 8, x[4] + 2*x[5] - 10, 2*x[6] + x[7] - 12, x[6] + 2*x[7] - 14, 2*x[8] + x[9] - 16, x[8] + 2*x[9] - 18])

#二階導數，也就是海森矩陣，冪次只有二次，所有二階導是常數

hess = lambda x: np.array([[2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2]])

if __name__ == "__main__":

x0 = np.array([1.0,1.0,1.0,3.0,1.0,3.0,1.0,1.0,2.0,3.0])

output=dampnm(fun, gfun, hess, x0)

print ("迭代次數:" output [2])

print ("近似最優解:" output [0])

print ("函數極小值點:" output [1])

print ("最終迭代誤差:" output [3])

3.擬牛頓法

import numpy as np

def bfgs(fun,gfun,x0):#這里只寫出了bfgs的實現

maxk = 1e5

rho = 0.55

sigma = 0.4

epsilon = 1e-5

k = 0

n = np.shape(x0)[0]

Bk = np.eye(n)#np.linalg.inv(hess(x0))

_epsilon=epsilon

while k < maxk:

gk = gfun(x0)

print(k,np.linalg.norm(gk))

if np.linalg.norm(gk) < epsilon:

_epsilon=np.linalg.norm(gk)

break

dk = -1.0*np.linalg.solve(Bk,gk)

m = 0

mk = 0

while m < 20:

if fun(x0+rho**m*dk) < fun(x0)+sigma*rho**m*np.dot(gk,dk):

mk = m

break

m += 1

#BFGS

x = x0 + rho**mk*dk

sk = x - x0

yk = gfun(x) - gk

if np.dot(sk,yk) > 0:

Bs = np.dot(Bk,sk)

ys = np.dot(yk,sk)

sBs = np.dot(np.dot(sk,Bk),sk)

Bk = Bk - 1.0*Bs.reshape((n,1))*Bs/sBs + 1.0*yk.reshape((n,1))*yk/ys

k += 1

x0 = x

return x0,fun(x0),k,_epsilon#

#目標函數

fun = lambda x: (x[0]-1)**2+(x[1]-1)**2+(x[2]-2)**2+(x[3]-3)**2+(x[4]-4)**2+(x[5]-5)**2+(x[6]-6)**2+(x[7]-7)**2+(x[8]-8)**2+(x[9]-9)**2+x[0]*x[1]+x[2]*x[3]+x[4]*x[5]+x[6]*x[7]+x[8]*x[9]

#一階導

gfun = lambda x: np.array([2*x[0] + x[1] - 2, x[0] + 2*x[1] - 2, 2*x[2] + x[3] - 4, x[2] + 2*x[3] - 6, 2*x[4] + x[5] - 8, x[4] + 2*x[5] - 10, 2*x[6] + x[7] - 12, x[6] + 2*x[7] - 14, 2*x[8] + x[9] - 16, x[8] + 2*x[9] - 18])

if __name__ == "__main__":

x0 = np.array([1.0,1.0,1.0,3.0,1.0,3.0,1.0,1.0,2.0,3.0])

output=bfgs(fun, gfun, x0)

print ("迭代次數:" output [2])

print ("近似最優解:" output [0])

print ("函數極小值點:" output [1])

print ("最終迭代誤差:" output [3])

這里是同一個目標函數在三種方法下的結果：近似最優解極小值迭代次數最終誤差

最速下降法[0.66665596 0.66665596 0.66665596 2.66665596 2.00003211 4.00003211 3.33440243 5.33247832 4.66724938 6.66628733]92.666 66 7892774127.0733410524

04242e-06

牛頓法[0.66666667 0.66666667 0.66666667 2.66666667 2. 4. 3.33333333 5.33333333 4.66666667 6.66666667]92.6666666666666710

擬牛頓法[0.66666652 0.66666652 0.66666652 2.66666652 2.00000043 4.00000043 3.33333781 5.33333172 4.66666955 6.6666665 ]92.6666666666905739.86253776

2278836e-06

如果不考慮計算二階導的支出，牛頓法還是比較生猛的。

完了-′???`。

如有興趣，可參考博客：最速下降法(梯度下降法)python實現_Big_Pai的博客-CSDN博客_最速下降法python?blog.csdn.net

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