改善算法——L-BFGS算法：一种优化算法的拟牛顿方法

#coding:UTF-8

from numpy import *
from function import *

def lbfgs(fun, gfun, x0):
    result = []#保留最终的结果
    maxk = 500#最大的迭代次数
    rho = 0.55
    sigma = 0.4
    
    H0 = eye(shape(x0)[0])
    
    #s和y用于保存最近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

#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

#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()