1 问题
对于下述一元函数：
$$
\phi(\alpha)=4(1024\alpha-4)4+(2\alpha-1)2 \tag{1}
$$
求取$\phi(\alpha)$在$[0,+\infty]$上的极小点$\alpha_0$。
2 分析
利用割线法的算法进行求解，在给定初值$\alpha{-1}$和$\alpha{0}$的情况下，迭代进行求解，求解算法如下：
$$
\alpha{k+1}=\frac{\phi'(x{k})x{(k-1)}-\phi'(x{k-1})x{(k)}}{\phi'(x{k})-\phi'(x^{(k-1)})} \tag{2}
$$
3 编程实现
利用Python进行实现：
import sys
def derivative(x, dx=1e-6):
# Computes the numerical derivative of a function.
return (g(x+dx) - g(x)) / dx
def g(x):
return 4*(1024*x-4)**4 + (2*x-1)**2
def secant_method(aList):
a = aList[-1]
b = aList[-2]
numerator = float(derivative(a)*b - derivative(b)*a)
denominator = float(derivative(a) - derivative(b))
step_length = numerator / denominator
if g(step_length) < g(aList[-1]):
aList.append(step_length)
secant_method(aList)
else:
return aList
aList = [1, 0.5]
finalList = secant_method(aList)
answer = finalList[-1]
print(answer)
得到结果：$\alpha_0=3.67\times10^{-3}$
4 遇到的问题
问题
最开始采用以下代码：
aList = [1, 0.5]
finalList = secant_method(aList)
answer = finalList[-1]
print(answer)
报错如下：
Traceback (most recent call last):
File "G:/PyCharm/PycharmProjects/Optimization/An introduction to optimization/diff.py", line 25, in <module>
answer = finalList[-1]
TypeError: 'NoneType' object is not subscriptable
Process finished with exit code 1
问题原因
报错原因在于列表被函数调用是一个in-place过程，将其赋给另一个对象时，传递的是一个None类型的数据
解决办法
修改为以下代码即可：
aList = [1, 0.5]
secant_method(aList)
answer = aList[-1]
print(answer)
