運籌學課上，首先介紹了非線性規劃算法中的無約束規劃算法。二分法和黃金分割法是屬于無約束規劃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)}

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)})

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：

1 importmath2 importnumpy as np3

4

5 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

6 return x*math.cos(x)7

8

9 Low = float(input("Please enter the lowbound:"))10 High = float(input("Please enter the highbound:"))11 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

12 echos = int(input("Please enter the echos:")) #迭代次數

13 small = float(input("Please enter the smallvalue:")) #公式中的Delta

14

15 for i in range(1, echos + 1):16 Lowvalue = anyfunction(0.5*(Low + High -small))17 Highvalue = anyfunction(0.5*(Low + High +small))18 print("echos:" +str(i))19 print(‘before‘ + "Lowbound:" + str(0.5*(Low + High - small)) + "Highbound:" + str(0.5*(Low + High +small)))20 print(‘Lowvalue:‘ + str(Lowvalue) + ‘ ‘ + ‘Highvalue:‘ +str(Highvalue))21 if(Lowvalue ==Highvalue):22 Low = 0.5*(Low + High -small)23 High = 0.5*(Low + High +small)24 elif(Lowvalue

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：

1 from math importsqrt, cos2 importnumpy as np3

4

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

6 return x*cos(x)7

8

9 Low = float(input("Please enter the lowbound:"))10 High = float(input("Please enter the highbound:"))11 High = np.pi #同上，使用時應該注釋掉

12 echos = int(input("Please enter the echos:"))13

14 #初始化，第一次運算不存在運算簡化

15 uniquevalue = ((sqrt(5)-1)/2)*(High-Low)16 value1 = anyfunction(High -uniquevalue)17 value2 = anyfunction(Low +uniquevalue)18

19 for i in range(1, echos + 1):20 print("echos:" +str(i))21 print(‘before‘ + "Lowbound:" + str(High - uniquevalue) + "Highbound:" + str(Low +uniquevalue))22 print(‘value1:‘ + str(value1) + ‘ ‘ + ‘value2:‘ +str(value2))23 #利用黃金分割法的性質減少一半的運算量

24 if(value1 ==value2):25 Low = High -uniquevalue26 High = Low +uniquevalue27 uniquevalue = ((sqrt(5)-1)/2)*(High-Low)28 value1 = anyfunction(High -uniquevalue)29 value2 = anyfunction(Low +uniquevalue)30 elif(value1

輸出結果如下:

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

原文：https://www.cnblogs.com/chester-cs/p/11751508.html

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