Python實現二分法和黃金分割法

　　運籌學課上，首先介紹了非線性規划算法中的無約束規划算法。二分法和黃金分割法是屬於無約束規划算法的一維搜索法中的代表。

　　二分法:$$x_{1}^{(k+1)}=\frac{1}{2}(x_{R}^{(k)}+x_{L}^{(k)}-\Delta)$$$$x_{2}^{(k+1)}=\frac{1}{2}(x_{R}^{(k)}+x_{L}^{(k)}+\Delta)$$

　　黃金分割法:$$x_{1}^{(k+1)}=x_{R}^{(k)}-(\frac{\sqrt{5}-1}{2})(x_{R}^{(k)}-x_{L}^{(k)})$$$$x_{2}^{(k+1)}=x_{L}^{(k)}+(\frac{\sqrt{5}-1}{2})(x_{R}^{(k)}-x_{L}^{(k)})$$

　　選擇的$x_{1}^{(k+1)}$和$x_{2}^{(k+1)}$一定滿足$$x_{L}^{(k)}<x_{1}^{(k+1)}<x_{2}^{(k+1)}<x_{R}^{(k)}$$

　　下面確定新的不確定空間$I^{(k+1)}$

　　情況1:若$f(x_{1}^{(k+1)})>f(x_{2}^{(k+1)})$,則$I^{(k+1)}=\left[x_{L}^{(k)},x_{2}^{(k+1)}\right]$

　　情況2:若$f(x_{1}^{(k+1)})<f(x_{2}^{(k+1)})$,則$I^{(k+1)}=\left[x_{1}^{(k+1)},x_{R}^{(k)}\right]$

　　情況3:若$f(x_{1}^{(k+1)})=f(x_{2}^{(k+1)})$,則$I^{(k+1)}=\left[x_{1}^{(k+1)},x_{2}^{(k+1)}\right]$

　　下面記錄下用Python實現二分法和黃金分割法的代碼。

　　二分法：

import math
import numpy as np


def anyfunction(x):  # 在這里我們定義任意一個指定初始區間內的單峰函數,以x*cos(x)為例
    return x*math.cos(x)


Low = float(input("Please enter the lowbound: "))
High = float(input("Please enter the highbound: "))
High = np.pi  # 在這里我們取初始上界為π,如果可以輸入則注釋掉這一行
echos = int(input("Please enter the echos: "))  # 迭代次數
small = float(input("Please enter the smallvalue: "))  # 公式中的Delta

for i in range(1, echos + 1):
    Lowvalue = anyfunction(0.5*(Low + High - small))
    Highvalue = anyfunction(0.5*(Low + High + small))
    print("echos: " + str(i))
    print('before ' + "Lowbound: " + str(0.5*(Low + High - small)) + " Highbound: " + str(0.5*(Low + High + small)))
    print('Lowvalue: ' + str(Lowvalue) + ' ' + 'Highvalue: ' + str(Highvalue))
    if(Lowvalue == Highvalue):
        Low = 0.5*(Low + High - small)
        High = 0.5*(Low + High + small)
    elif(Lowvalue < Highvalue):
        Low = 0.5*(Low + High - small)
    else:
        High = 0.5*(Low + High + small)
    print("Lowbound: " + str(Low) + " Highbound: " + str(High))

　　輸出結果如下:

　　5次循環后極值點被限制在[0.7828981633974482,0.8907604338221292]內。

　　黃金分割法：

from math import sqrt, cos
import numpy as np


def anyfunction(x):  # 同上以函數x*cos(x)為例
    return x*cos(x)


Low = float(input("Please enter the lowbound: "))
High = float(input("Please enter the highbound: "))
High = np.pi  # 同上，使用時應該注釋掉
echos = int(input("Please enter the echos: "))

# 初始化，第一次運算不存在運算簡化
uniquevalue = ((sqrt(5)-1)/2)*(High-Low)
value1 = anyfunction(High - uniquevalue)
value2 = anyfunction(Low + uniquevalue)

for i in range(1, echos + 1):
    print("echos: " + str(i))
    print('before ' + "Lowbound: " + str(High - uniquevalue) + " Highbound: " + str(Low + uniquevalue))
    print('value1: ' + str(value1) + ' ' + 'value2: ' + str(value2))
    # 利用黃金分割法的性質減少一半的運算量
    if(value1 == value2):
        Low = High - uniquevalue
        High = Low + uniquevalue
        uniquevalue = ((sqrt(5)-1)/2)*(High-Low)
        value1 = anyfunction(High - uniquevalue)
        value2 = anyfunction(Low + uniquevalue)
    elif(value1 < value2):
        Low = High - uniquevalue
        uniquevalue = ((sqrt(5)-1)/2)*(High-Low)
        value1 = value2
        value2 = anyfunction(Low + uniquevalue)
    else:
        High = Low + uniquevalue
        uniquevalue = ((sqrt(5)-1)/2)*(High-Low)
        value2 = value1
        value1 = anyfunction(High - uniquevalue)
    print("Lowbound: " + str(Low) + " Highbound: " + str(High))

　　輸出結果如下:

 　　5次循環后極值點被限制在[0.7416294238611398,1.0249066567190932]