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一维搜索之黄金分割法

发布于2020-04-23 19:30 阅读(827) 评论(0) 点赞(0) 收藏(3)

黄金分割法（0.618法）优化的原理

例题

这里用黄金分割法求解 $y = x^2-7x+10$ 的最优解,，初始区间 $[a,b]=[2,8]$ , 收敛精度为 0.001。

$k$	$a$	$x_{k}^{1}$	$x_{k}^{2}$	$b$	$f(x_{k}^{1})$	$f(x_{k}^{2})$	$[a,b]$	b-a
1	2	–	–	8	–	–	$[2,8]$	6
$2$	$2$	$4.292$	$5.708$	$8$	$-1.6227359999999997$	$2.6252640000000014$	$[2, 5.708]$	$3.708$
$3$	$2$	$3.416456$	$4.291544$	$5.708$	$-2.2430204000639993$	$-1.6234580960639988$	$[2, 4.291544 ]$	$2.291544$
4	$2.875369808$	$3.416348349344$	$3.750565458656$	$4.291544$	$-2.243002401342528$	$-2.2429843830315406$	$[ 2.875369808 , 3.750565458656]$	$0.8751956506560004$
–	–	–	–	–	–	–	–	–
18	$3.498963664784351$	$3.4996049019558986$	$3.5000010589519333$	$3.500642296123481$	$-2.2499998438975357$	$-2.2499999999988773$	$[ 3.4996049019558986 , 3.500642296123481 ]$	$0.0010373941675823062$
19	$3.4996049019558986$	$3.500001186527915$	$3.5002460115514644$	$3.500642296123481$	$-2.249999999998593$	$-2.2499999394783163$	$[ 3.4996049019558986 , 3.5002460115514644 ]$	$0.004395197314465804$

代码实现

import matplotlib.pyplot as plt
import numpy as np
from pylab import *

#目标函数
def function(x):
    y = x**2 - 7 * x + 10
    return y
#黄金分割法
def GoldenBorder(a, b, accuracy):
    x1 = a + 0.382*(b - a)
    x2 = a + 0.618*(b - a)
    fx1 = function(x1)
    fx2 = function(x2)
    print( "a=", a,"x1=", x1, "x2=", x2 , ",b=", b,",f(x1)=",fx1, ",f(x2)=", fx2,  )
    if fx1 < fx2:
        a = a
        b = x2
    else:
        a = x1
        b = b
    accuracy = b - a
    print("NewRange[", a, ", ",b,"]","accuracy=", b-a)
    return a, b ,accuracy

a = 2
b = 8
value = []
accuracy = b - a
epoch = 0
accuracies = []
ys = []
epoches=[]
xs = []
while accuracy > 0.001:
    epoch += 1
    print("-"*100)
    print("Epoch：",epoch)
    a, b, accuracy = GoldenBorder(a, b, accuracy)
    accuracies.append(accuracy)
    ys.append(function( a + (b - a) / 2 ))
    epoches.append(epoch)
    xs.append(a + (b - a) / 2)
x = a + (b - a) / 2    
y = function(x)     
print("-"*100)
print("最优点：x*=", x,",f(x*)=", y)

plt.title(' Golden Section Search ')

plt.subplot(2,1,1)
plt.plot(epoches,accuracies, ":o", label = 'accuracy', color="blue")
xticks(np.linspace(0,19,20,endpoint=True))
plt.ylabel('f(x)')
plt.legend()

plt.subplot(2,1,2)
plt.plot(ys, "-.o", label = 'f(x)', color="red")
xticks(np.linspace(0,19,20,endpoint=True))
plt.xlabel('epoch')
plt.ylabel('accuracy')
plt.legend()
plt.show()

在这里插入图片描述

X = np.linspace(3, 4, 20)
Epoch = np.linspace(0, 19,20)
X, Epoch= np.meshgrid(X, Epoch)
Y = X**2 - X * 7 + 10

fig = plt.figure(figsize = ((8,6)))
ax = fig.gca(projection='3d')
ax.plot_surface(Epoch ,X , Y, color="yellow", alpha=0.5)
ax.plot(epoches ,xs , ys,"-.o",color="blue")
xticks(np.linspace(0,19,20,endpoint=True))
ax.set_xlabel('epoch')                #设置坐标轴标签
ax.set_ylabel('x')
ax.set_zlabel(' f(x) ')
plt.show()