下面是关于“Python优化算法之遗传算法案例代码”的完整攻略。
1. 遗传算法简介
遗传算法是一种基于自然选择和遗传学原理的优化算法，它通过模拟生物进化过程，从而实现对问题的优化。遗传算法的基本流程包括初始化种群、选择、交叉、变异等步骤。
2. Python实现遗传算法
2.1 初始化种群
在遗传算法中，种群是指一组个体，每个个体都代表了问题的一个解。在Python中，我们可以使用 numpy 库生成随机的二进制编码来初始化种群。
下面是一个初始化种群的示例：
import numpy as np
def init_population(pop_size, chrom_length):
population = np.random.randint(2, size=(pop_size, chrom_length))
return population
在这个示例中，我们使用 numpy.random.randint() 函数生成一个大小为 (pop_size, chrom_length) 的随机矩阵，其中每个元素都是 0 或 1。
2.2 选择
选择是指从种群中选择一部分个体作为下一代种群的父代。在遗传算法中，选择通常使用轮盘赌选择或锦标赛选择。
下面是一个使用轮盘赌选择的示例：
def roulette_selection(population, fitness):
fitness_sum = np.sum(fitness)
fitness_ratio = fitness / fitness_sum
cum_sum = np.cumsum(fitness_ratio)
selected_index = []
for i in range(len(population)):
r = np.random.rand()
for j in range(len(cum_sum)):
if r < cum_sum[j]:
selected_index.append(j)
break
selected_population = population[selected_index]
return selected_population
在这个示例中，我们首先计算每个个体的适应度比例，然后计算适应度比例的累积和。接着，我们使用 numpy.random.rand() 函数生成一个随机数，然后根据随机数选择一个个体。重复这个过程，直到选择足够数量的个体。最后，我们返回选择的个体。
2.3 交叉
交叉是指将两个父代个体的染色体进行配对，生成新的子代个体。在遗传算法中，交叉通常使用单点交叉或多点交叉。
下面是一个使用单点交叉的示例：
def single_point_crossover(parent1, parent2):
chrom_length = len(parent1)
crossover_point = np.random.randint(1, chrom_length)
child1 = np.concatenate((parent1[:crossover_point], parent2[crossover_point:]))
child2 = np.concatenate((parent2[:crossover_point], parent1[crossover_point:]))
return child1, child2
在这个示例中，我们首先生成一个随机数，作为交叉点。然后，我们将两个父代个体的染色体在交叉点处进行切割，并将切割后的部分进行交换，生成两个新的子代个体。
2.4 变异
变异是指对个体的染色体进行随机的改变，从而生成新的个体。在遗传算法中，变异通常使用位翻转或随机重置。
下面是一个使用位翻转的示例：
def bit_flip_mutation(individual, mutation_rate):
chrom_length = len(individual)
for i in range(chrom_length):
if np.random.rand() < mutation_rate:
individual[i] = 1 - individual[i]
return individual
在这个示例中，我们遍历个体的每个基因，如果随机数小于变异率，则将该基因进行翻转。
3. 示例说明
下面是一个使用遗传算法求解函数最小值的示例：
import numpy as np
def fitness_func(x):
return np.sin(10 * np.pi * x) / (2 * x) + (x - 1) ** 4
def init_population(pop_size, chrom_length):
population = np.random.randint(2, size=(pop_size, chrom_length))
return population
def roulette_selection(population, fitness):
fitness_sum = np.sum(fitness)
fitness_ratio = fitness / fitness_sum
cum_sum = np.cumsum(fitness_ratio)
selected_index = []
for i in range(len(population)):
r = np.random.rand()
for j in range(len(cum_sum)):
if r < cum_sum[j]:
selected_index.append(j)
break
selected_population = population[selected_index]
return selected_population
def single_point_crossover(parent1, parent2):
chrom_length = len(parent1)
crossover_point = np.random.randint(1, chrom_length)
child1 = np.concatenate((parent1[:crossover_point], parent2[crossover_point:]))
child2 = np.concatenate((parent2[:crossover_point], parent1[crossover_point:]))
return child1, child2
def bit_flip_mutation(individual, mutation_rate):
chrom_length = len(individual)
for i in range(chrom_length):
if np.random.rand() < mutation_rate:
individual[i] = 1 - individual[i]
return individual
def genetic_algorithm(pop_size, chrom_length, max_iter, mutation_rate):
population = init_population(pop_size, chrom_length)
for i in range(max_iter):
fitness = np.array([fitness_func(x) for x in population])
selected_population = roulette_selection(population, fitness)
new_population = []
for j in range(pop_size // 2):
parent1 = selected_population[j * 2]
parent2 = selected_population[j * 2 + 1]
child1, child2 = single_point_crossover(parent1, parent2)
child1 = bit_flip_mutation(child1, mutation_rate)
child2 = bit_flip_mutation(child2, mutation_rate)
new_population.append(child1)
new_population.append(child2)
population = np.array(new_population)
fitness = np.array([fitness_func(x) for x in population])
best_index = np.argmin(fitness)
best_individual = population[best_index]
best_fitness = fitness[best_index]
return best_individual, best_fitness
best_individual, best_fitness = genetic_algorithm(pop_size=100, chrom_length=20, max_iter=100, mutation_rate=0.01)
print("Best individual:", best_individual)
print("Best fitness:", best_fitness)
在这个示例中，我们定义了一个函数 fitness_func(x)，用于计算函数的适应度。然后，我们使用 init_population() 函数初始化种群。接着，我们使用 roulette_selection() 函数进行选择，使用 single_point_crossover() 函数进行交叉，使用 bit_flip_mutation() 函数进行变异。最后，我们使用 genetic_algorithm() 函数进行遗传算法求解最小值。
3.2 使用遗传算法求解TSP问题
下面是一个使用遗传算法求解TSP问题的示例：
import numpy as np
import matplotlib.pyplot as plt
def init_population(pop_size, city_num):
population = np.zeros((pop_size, city_num), dtype=int)
for i in range(pop_size):
population[i] = np.random.permutation(city_num)
return population
def calc_distance(city1, city2):
return np.sqrt(np.sum((city1 - city2) ** 2))
def calc_fitness(individual, cities):
fitness = 0
for i in range(len(individual) - 1):
city1 = cities[individual[i]]
city2 = cities[individual[i + 1]]
fitness += calc_distance(city1, city2)
city1 = cities[individual[-1]]
city2 = cities[individual[0]]
fitness += calc_distance(city1, city2)
return 1 / fitness
def roulette_selection(population, fitness):
fitness_sum = np.sum(fitness)
fitness_ratio = fitness / fitness_sum
cum_sum = np.cumsum(fitness_ratio)
selected_index = []
for i in range(len(population)):
r = np.random.rand()
for j in range(len(cum_sum)):
if r < cum_sum[j]:
selected_index.append(j)
break
selected_population = population[selected_index]
return selected_population
def order_crossover(parent1, parent2):
chrom_length = len(parent1)
crossover_point1 = np.random.randint(1, chrom_length - 1)
crossover_point2 = np.random.randint(crossover_point1 + 1, chrom_length)
child1 = np.zeros(chrom_length, dtype=int)
child2 = np.zeros(chrom_length, dtype=int)
child1[crossover_point1:crossover_point2] = parent1[crossover_point1:crossover_point2]
child2[crossover_point1:crossover_point2] = parent2[crossover_point1:crossover_point2]
for i in range(crossover_point2, chrom_length):
for j in range(chrom_length):
if parent1[i] not in child2:
child2[j] = parent1[i]
break
if parent2[i] not in child1:
child1[j] = parent2[i]
break
for i in range(crossover_point1):
for j in range(chrom_length):
if parent1[i] not in child2:
child2[j] = parent1[i]
break
if parent2[i] not in child1:
child1[j] = parent2[i]
break
return child1, child2
def swap_mutation(individual, mutation_rate):
chrom_length = len(individual)
for i in range(chrom_length):
if np.random.rand() < mutation_rate:
j = np.random.randint(chrom_length)
individual[i], individual[j] = individual[j], individual[i]
return individual
def genetic_algorithm(pop_size, city_num, max_iter, mutation_rate, cities):
population = init_population(pop_size, city_num)
best_fitness_list = []
for i in range(max_iter):
fitness = np.array([calc_fitness(individual, cities) for individual in population])
best_fitness = np.max(fitness)
best_fitness_list.append(best_fitness)
selected_population = roulette_selection(population, fitness)
new_population = []
for j in range(pop_size // 2):
parent1 = selected_population[j * 2]
parent2 = selected_population[j * 2 + 1]
child1, child2 = order_crossover(parent1, parent2)
child1 = swap_mutation(child1, mutation_rate)
child2 = swap_mutation(child2, mutation_rate)
new_population.append(child1)
new_population.append(child2)
population = np.array(new_population)
fitness = np.array([calc_fitness(individual, cities) for individual in population])
best_index = np.argmax(fitness)
best_individual = population[best_index]
best_fitness = fitness[best_index]
return best_individual, best_fitness, best_fitness_list
def plot_tsp(best_individual, cities):
plt.plot(cities[:, 0], cities[:, 1], 'o')
for i in range(len(best_individual) - 1):
city1 = cities[best_individual[i]]
city2 = cities[best_individual[i + 1]]
plt.plot([city1[0], city2[0]], [city1[1], city2[1]], 'k-')
city1 = cities[best_individual[-1]]
city2 = cities[best_individual[0]]
plt.plot([city1[0], city2[0]], [city1[1], city2[1]], 'k-')
plt.show()
cities = np.random.rand(20, 2)
best_individual, best_fitness, best_fitness_list = genetic_algorithm(pop_size=100, city_num=20, max_iter=1000, mutation_rate=0.01, cities=cities)
print("Best individual:", best_individual)
print("Best fitness:", best_fitness)
plot_tsp(best_individual, cities)
在这个示例中，我们首先定义了一个函数 calc_distance(city1, city2)，用于计算两个城市之间的距离。然后，我们使用 init_population() 函数初始化种群。接着，我们使用 calc_fitness() 函数计算个体的适应度，使用 roulette_selection() 函数进行选择，使用 order_crossover() 函数进行交叉，使用 swap_mutation() 函数进行变异。最后，我们使用 genetic_algorithm() 函数进行遗传算法求解TSP问题，并使用 plot_tsp() 函数绘制最优路径。
4. 说明
Python中提供了多种优化算法，包括遗传算法、粒子群算法、蚁群算法等。这些算法可以帮助我们对问题进行优化，从而实现对数据的分析和预测。在使用这些算法时，我们需要根据具体的问题选择合适的算法，并据模型的特点和数据集的特征进行调参。