一、BFGS算法
在“优化算法——拟牛顿法之BFGS算法”中，我们得到了BFGS算法的校正公式：
利用Sherman-Morrison公式可对上式进行变换，得到
令，则得到：
二、BGFS算法存在的问题
在BFGS算法中。每次都要存储近似Hesse矩阵
B_k^{-1}" title="B_k^{-1}" alt="" />，在高维数据时，存储浪费非常多的存储空间，而在实际的运算过程中。我们须要的是搜索方向。因此出现了L-BFGS算法。是对BFGS算法的一种改进算法。
在L-BFGS算法中。仅仅保存近期的次迭代信息。以减少数据的存储空间。
三、L-BFGS算法思路
令
\rho&space;_k=\frac{1}{y_k^Ts_k}" title="\rho _k=\frac{1}{y_k^Ts_k}" alt="" />。
V_k=I-\frac{y_ks_k^T}{y_k^Ts_k}" title="V_k=I-\frac{y_ks_k^T}{y_k^Ts_k}" alt="" />，则BFGS算法中的
H_{k+1}" title="H_{k+1}" alt="" />能够表示为：
若在初始时，假定初始的矩阵
H_0=I" title="H_0=I" alt="" />，则我们能够得到：
H_{1}=V_0^TH_0V_0+\rho&space;_0s_0s_0^T" title="H_{1}=V_0^TH_0V_0+\rho _0s_0s_0^T" alt="" />
\begin{align*}&space;H_2&space;&=&space;V_1^TH_1V_1+\rho&space;_1s_1s_1^T\\&space;&=&space;V_1^T\left&space;(&space;V_0^TH_0V_0+\rho&space;_0s_0s_0^T&space;\right&space;)V_1+\rho&space;_1s_1s_1^T\\&space;&=&space;V_1^TV_0^TH_0V_0V_1+V_1^T\rho&space;_0s_0s_0^TV_1+\rho&space;_1s_1s_1^T&space;\end{align*}" title="\begin{align*} H_2 &= V_1^TH_1V_1+\rho _1s_1s_1^T\\ &= V_1^T\left ( V_0^TH_0V_0+\rho _0s_0s_0^T \right )V_1+\rho _1s_1s_1^T\\ &= V_1^TV_0^TH_0V_0V_1+V_1^T\rho _0s_0s_0^TV_1+\rho _1s_1s_1^T \end{align*}" alt="" />
若此时。仅仅保留近期的步：
\begin{align*}&space;H_{k+1}&space;&=&space;\left&space;(&space;V_k^TV_{k-1}^T\cdots&space;V_{k-m}^T&space;\right&space;)H_0\left&space;(&space;V_{k-m}\cdots&space;V_{k-1}V_k&space;\right&space;)\\&space;&+&space;\left&space;(&space;V_k^TV_{k-1}^T\cdots&space;V_{k-m}^T&space;\right&space;)\rho&space;_1s_1s_1^T\left&space;(&space;V_{k-m}\cdots&space;V_{k-1}V_k&space;\right&space;)\\&space;&+&space;\cdots&space;\\&space;&+&space;V_k^T\rho&space;_{k-1}s_{k-1}s_{k-1}^TV_k\\&space;&+&space;\rho&space;_ks_ks_k^T&space;\end{align*}" title="\begin{align*} H_{k+1} &= \left ( V_k^TV_{k-1}^T\cdots V_{k-m}^T \right )H_0\left ( V_{k-m}\cdots V_{k-1}V_k \right )\\ &+ \left ( V_k^TV_{k-1}^T\cdots V_{k-m}^T \right )\rho _1s_1s_1^T\left ( V_{k-m}\cdots V_{k-1}V_k \right )\\ &+ \cdots \\ &+ V_k^T\rho _{k-1}s_{k-1}s_{k-1}^TV_k\\ &+ \rho _ks_ks_k^T \end{align*}" alt="" />
这样在L-BFGS算法中。不再保存完整的
H_k" title="H_k" alt="" />。而是存储向量序列和
。须要矩阵
时，使用向量序列
\left&space;\{&space;s_k&space;\right&space;\}" title="\left \{ s_k \right \}" alt="" style="font-family:KaiTi_GB2312; font-size:18px" />和计算就能够得到。而向量序列
和
也不是全部都要保存，仅仅要保存最新的
步向量就可以。
四、L-BFGS算法中的方向的计算方法
五、实验仿真
lbfgs.py
#coding:UTF-8 from numpy import *
from function import * def lbfgs(fun, gfun, x0):
result = []#保留终于的结果
maxk = 500#最大的迭代次数
rho = 0.55
sigma = 0.4 H0 = eye(shape(x0)[0]) #s和y用于保存近期m个，这里m取6
s = []
y = []
m = 6 k = 1
gk = mat(gfun(x0))#计算梯度
dk = -H0 * gk
while (k < maxk):
n = 0
mk = 0
gk = mat(gfun(x0))#计算梯度
while (n < 20):
newf = fun(x0 + rho ** n * dk)
oldf = fun(x0)
if (newf < oldf + sigma * (rho ** n) * (gk.T * dk)[0, 0]):
mk = n
break
n = n + 1 #LBFGS校正
x = x0 + rho ** mk * dk
#print x #保留m个
if k > m:
s.pop(0)
y.pop(0) #计算最新的
sk = x - x0
yk = gfun(x) - gk s.append(sk)
y.append(yk) #two-loop的过程
t = len(s)
qk = gfun(x)
a = []
for i in xrange(t):
alpha = (s[t - i - 1].T * qk) / (y[t - i - 1].T * s[t - i - 1])
qk = qk - alpha[0, 0] * y[t - i - 1]
a.append(alpha[0, 0])
r = H0 * qk for i in xrange(t):
beta = (y[i].T * r) / (y[i].T * s[i])
r = r + s[i] * (a[t - i - 1] - beta[0, 0]) if (yk.T * sk > 0):
dk = -r k = k + 1
x0 = x
result.append(fun(x0)) return result
function.py
#coding:UTF-8
'''
Created on 2015年5月19日 @author: zhaozhiyong
''' from numpy import * #fun
def fun(x):
return 100 * (x[0,0] ** 2 - x[1,0]) ** 2 + (x[0,0] - 1) ** 2 #gfun
def gfun(x):
result = zeros((2, 1))
result[0, 0] = 400 * x[0,0] * (x[0,0] ** 2 - x[1,0]) + 2 * (x[0,0] - 1)
result[1, 0] = -200 * (x[0,0] ** 2 - x[1,0])
return result
testLBFGS.py
#coding:UTF-8
'''
Created on 2015年6月6日 @author: zhaozhiyong
''' from lbfgs import * import matplotlib.pyplot as plt x0 = mat([[-1.2], [1]])
result = lbfgs(fun, gfun, x0)
print result n = len(result)
ax = plt.figure().add_subplot(111)
x = arange(0, n, 1)
y = result
ax.plot(x,y) plt.show()
实验结果
watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvZ29vZ2xlMTk4OTAxMDI=/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" alt="" />