给出CMA-ES代码,如何实现协方差矩阵自适应性?和DE算法的区别?
时间: 2023-12-03 12:03:46 浏览: 28
以下是一个基于Python实现的CMA-ES算法示例代码,其中包括协方差矩阵自适应性的实现部分:
```python
numpy as np
from numpy.random import randn
def cma_es(fitness_function, initial_mean, initial_sigma, max_iterations, population_size):
n = len(initial_mean)
x_mean = initial_mean
sigma = initial_sigma
C = np.identity(n)
p_sigma = np.zeros(n)
p_c = np.zeros(n)
weights = np.log(population_size/2+1) - np.log(np.arange(1,population_size+1))
mu_eff = np.sum(weights)**2 / np.sum(weights**2)
c_sigma = (mu_eff + 2) / (n + mu_eff + 5)
c_c = 4 / (n + 4)
D = 1 + 2*np.max([0, np.sqrt((mu_eff-1)/(n+1))-1]) + c_sigma
eigeneval = 0
x = np.zeros((n,population_size))
x = x_mean.reshape((-1,1)) + sigma * np.dot(C, randn(n,population_size))
fitness = np.zeros(population_size)
for i in range(population_size):
fitness[i] = fitness_function(x[:,i])
eigeneval += population_size
idx = np.argsort(fitness)
x_old = x_mean
x_mean = np.dot(x[:,idx[:int(population_size/2)]], weights)
p_sigma = (1 - c_sigma) * p_sigma + np.sqrt(c_sigma * (2-c_sigma) * mu_eff) * np.dot(np.linalg.inv(C), (x_mean-x_old)/sigma)
h_sigma = 1 if np.linalg.norm(p_sigma)/np.sqrt(1-(1-c_sigma)**(2*(eigeneval/population_size))) < (1.4 + 2/(n+1)) else 0
sigma = sigma * np.exp((c_sigma / D) * (np.linalg.norm(p_sigma)/np.sqrt(1-(1-c_sigma)**(2*(eigeneval/population_size))) - 1))
if h_sigma == 1:
C = np.identity(n)
else:
y = np.dot(np.linalg.inv(C), (x[:,idx[:int(population_size/2)]]-x_old)/sigma)
p_c = (1 - c_c) * p_c + np.sqrt(c_c * (2-c_c) * mu_eff) * np.dot(weights, y.T)
C = ((1 - c1 - c_mu) * C + c1 * np.outer(p_c,p_c) + c_mu * np.dot(np.dot(y,np.diag(weights)),y.T)) + (1-h_sigma) * c_sigma*(2-c_sigma) * C
eigeneval += 2*population_size
return x[:,idx[0]], fitness[idx[0]], eigeneval
```
在CMA-ES中,协方差矩阵自适应性是通过调整协方差矩阵C来实现的。具体来说,C的更新公式包括两个部分:一是通过历史个体演化方向y的自适应更新,二是通过历史步长的自适应更新。其中,历史个体演化方向y的自适应更新主要体现在C的加权平均部分,而历史步长的自适应更新主要体现在C的系数部分。
相比之下,DE算法没有明确的协方差矩阵自适应性实现。DE算法主要通过选择性地使用不同的变异策略来实现自适应性,例如控制参数F和CR的变异策略、自适应差分进化策略等。此外,DE算法中的种群个体可以进行交叉和变异操作,而CMA-ES中的种群个体主要通过个体演化方向y和步长sigma来进行优化。