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| import numpy as np
# import tensorflow as tf
def gfun(x): # 梯度
# x = tf.Variable(x, dtype=tf.float32)
# with tf.GradientTape() as tape:
# tape.watch(x)
# z = fun(x)
# return tape.gradient(z, x).numpy() # 这里使用TensorFlow来求梯度，直接手算梯度返回也行
return np.array([4 * (x[0] - 2) ** 3 + 2 * (x[0] - 2 * x[1]), -4 * (x[0] - 2 * x[1])]).reshape(len(x), 1)
def fun(x): # 函数
return (x[0] - 2) ** 4 + (x[0] - 2 * x[1]) ** 2
def bfgs_armijo(x0):
'''秩1的基于armijo搜索的拟牛顿算法'''
maxk = 5000
rho = .55
sigma = .4
epsilon = 1e-5
k = 0
n = len(x0)
Hk = np.eye(n)
while k < maxk:
gk = gfun(x0)
if np.linalg.norm(gk) < epsilon:
break
dk = -Hk @ gk
m = 0
mk = 0
while m < 20: # 使用Armijo搜索（非精确线搜索）
if fun(x0 + rho ** m * dk) < fun(x0) + sigma * rho ** m * gk.T @ dk:
mk = m
break
m += 1
x = x0 + rho ** mk * dk
sk = x - x0
yk = gfun(x) - gk
Hk = Hk + (sk - Hk @ yk) @ (sk - Hk @ yk).T / ((sk - Hk @ yk).T @ yk)
k += 1
x0 = x
return x0, fun(x0), k
def bfgs_wolfe(x0):
'''基于wolfe搜索的秩1的拟牛顿算法'''
maxk = 5000
epsilon = 1e-5
k = 0
n = len(x0)
Hk = np.eye(n)
while k < maxk:
gk = gfun(x0)
if np.linalg.norm(gk) < epsilon:
break
dk = -Hk @ gk
rho = 0.4
sigma = 0.5
a = 0
b = np.inf
alpha = 1
while True: # 使用Wolfe搜索
if not ((fun(x0) - fun(x0 + alpha * dk)) >= (-rho * alpha * gfun(x0).T @ dk)):
b = alpha
alpha = (a + alpha) / 2
continue
if not (gfun(x0 + alpha * dk).T @ dk >= sigma * gfun(x0).T @ dk):
a = alpha
alpha = np.min([2 * alpha, (alpha + b) / 2])
continue
break
x = x0 + alpha * dk
sk = x - x0
yk = gfun(x) - gk
Hk = Hk + (sk - Hk @ yk) @ (sk - Hk @ yk).T / ((sk - Hk @ yk).T @ yk)
k += 1
x0 = x
return x0, fun(x0), k
def dfp_wolfe(x0):
'''基于armijo搜索的秩2的拟牛顿法
当采用精确线搜索时，矩阵序列Hk的正定性条件sk.T@yk>0可以被满足
一般来说，armijo准则不能满足这一条件需要修正一个条件：yk.T@sk>0
'''
maxk = 5000
epsilon = 1e-5
k = 0
n = len(x0)
Hk = np.eye(n) # 初始化Hk为单位阵
while k < maxk:
gk = gfun(x0)
if np.linalg.norm(gk) < epsilon: # 迭代结束条件
break
dk = -Hk @ gk
rho = 0.4
sigma = 0.5
a = 0
b = np.inf
alpha = 1
while True:
if not ((fun(x0) - fun(x0 + alpha * dk)) >= (-rho * alpha * gfun(x0).T @ dk)):
b = alpha
alpha = (a + alpha) / 2
continue
if not (gfun(x0 + alpha * dk).T @ dk >= sigma * gfun(x0).T @ dk):
a = alpha
alpha = np.min([2 * alpha, (alpha + b) / 2])
continue
break
x = x0 + alpha * dk
sk = x - x0
yk = gfun(x) - gk
Hk = Hk - (Hk @ yk @ yk.T @ Hk) / (yk.T @ Hk @ yk) + (sk @ sk.T) / (sk.T @ yk)
k += 1
x0 = x
return x0, fun(x0), k
def dfp_armijo(x0):
'''基于armijo搜索的秩2的拟牛顿法
当采用精确线搜索时，矩阵序列Hk的正定性条件sk.T@yk>0可以被满足
一般来说，armijo准则不能满足这一条件需要修正一个条件：yk.T@sk>0
'''
maxk = 5000
rho = .55
sigma = .4
epsilon = 1e-5
k = 0
n = len(x0)
Hk = np.eye(n)
while k < maxk:
gk = gfun(x0)
if np.linalg.norm(gk) < epsilon:
break
dk = -Hk @ gk
m = 0
mk = 0
while m < 20: # 使用Armijo搜索（非精确线搜索）
if fun(x0 + rho ** m * dk) < fun(x0) + sigma * rho ** m * gk.T @ dk:
mk = m
break
m += 1
x = x0 + rho ** mk * dk
sk = x - x0
yk = gfun(x) - gk
if sk.T @ yk > 0:
Hk = Hk - (Hk @ yk @ yk.T @ Hk) / (yk.T @ Hk @ yk) + (sk @ sk.T) / (sk.T @ yk)
k += 1
x0 = x
return x0, fun(x0), k
if __name__ == '__main__':
x0 = np.array([[0], [0]])
x0, val, k = bfgs_armijo(x0) # BFGS使用armijo准则
print(f'近似最优点：\n{x0}\n迭代次数：{k}\n目标函数值：{val.item()}')
x0 = np.array([[0], [0]])
x0, val, k = bfgs_wolfe(x0) # BFGS使用wolfe准则
print(f'近似最优点：\n{x0}\n迭代次数：{k}\n目标函数值：{val.item()}')
x0 = np.array([[0], [0]])
x0, val, k = dfp_armijo(x0) # DFP使用armijo准则
print(f'近似最优点：\n{x0}\n迭代次数：{k}\n目标函数值：{val.item()}')
x0 = np.array([[0], [0]])
x0, val, k = dfp_wolfe(x0) # DFP使用wolfe准则
print(f'近似最优点：\n{x0}\n迭代次数：{k}\n目标函数值：{val.item()}')
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