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【Python】人工智能-机器学习——不调库手撕演化算法解决函数最小值问题

1 作业内容描述

1.1 背景

  1. 现在有一个函数 3 − s i n 2 ( j x 1 ) − s i n 2 ( j x 2 ) 3-sin^2(jx_1)-sin^2(jx_2) 3−sin2(jx1​)−sin2(jx2​),有两个变量 x 1 x_1 x1​ 和 x 2 x_2 x2​,它们的定义域为 x 1 , x 2 ∈ [ 0 , 6 ] x_1,x_2\in[0,6] x1​,x2​∈[0,6],并且 j = 2 j=2 j=2,对于此例,所致对于 j = 2 , 3 , 4 , 5 j=2,3,4,5 j=2,3,4,5分别有 16,36,64,100 个全局最优解。
  2. 现在有一个Shubert函数 ∏ i = 1 n ∑ j = 1 5 j cos ⁡ [ ( j + 1 ) x i + j ] \prod_{i=1}^{n}\sum_{j=1}^{5}j\cos[(j+1)x_i+j] ∏i=1n​∑j=15​jcos[(j+1)xi​+j],其中定义域为 − 10 < x i < 10 -10<x_i<10 −10<xi​<10,对于此问题,当n=2时有18个不同的全局最优解

1.2 要求

  1. 求该函数的最小值即 m i n ( 3 − s i n 2 ( j x 1 ) − s i n 2 ( j x 2 ) ) min(3-sin^2(jx_1)-sin^2(jx_2)) min(3−sin2(jx1​)−sin2(jx2​)),j=2,精确到小数点后6位。
  2. 求该Shubert函数的最小值即 m i n ( ∏ i = 1 2 ∑ j = 1 5 j cos ⁡ [ ( j + 1 ) x i + j ] ) min(\prod_{i=1}^{2}\sum_{j=1}^{5}j\cos[(j+1)x_i+j]) min(∏i=12​∑j=15​jcos[(j+1)xi​+j]),精确到小数点后6位

2 作业已完成部分和未完成部分

该作业已经全部完成,没有未完成的部分。
Value在这里插入图片描述Colab NotebookGithub Rep

3. 作业运行结果截图

最后跑出的结果如下:

  1. 第一个函数的最小值为 1.0000000569262162
  2. 第二个函数的最小值为-186.73042323192567

4 核心代码和步骤

4.1 基本的步骤

  1. 定义目标函数 objective_function:使用了一个二维的目标函数,即 3 − s i n 2 ( j x 1 ) − s i n 2 ( j x 2 ) 3-sin^2(jx_1)-sin^2(jx_2) 3−sin2(jx1​)−sin2(jx2​)。
  2. 定义选择函数 crossover:用于交叉操作,通过交叉率(crossover_rate)确定需要进行交

叉的父母对的数量,并在这些父母对中交换某些变量的值。

  1. 定义变异函数 mutate:用于变异操作,通过变异率(mutation_rate)确定需要进行变异

的父母对的数量,并在这些父母对中随机改变某些变量的值。

  1. 定义进化算法 evolutionary_algorithm:初始化种群,其中每个个体都是一个二维向量。在

每一代中,计算每个个体的适应度值,绘制三维图表展示种群分布和最佳解。

  1. 更新全局最佳解。根据适应度值确定复制的数量并形成繁殖池。选择父母、进行交叉和变

异,更新种群。重复上述步骤直到达到指定的迭代次数。

  1. 设置算法参数:population_size:种群大小。;num_generations:迭代的次数。;muta

tion_rate:变异率。;crossover_rate:交叉率。

  1. 运行进化算法 evolutionary_algorithm:调用进化算法函数并获得最终的最佳解、最佳适

应度值和每一代的演化数据。

  1. 输出结果:打印最终的最佳解和最佳适应度值。输出每个迭代步骤的最佳适应度值。
  2. 可视化结果:绘制函数曲面和最优解的三维图表。绘制适应度值随迭代次数的变化曲线。

4.2 第一个函数

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3−sin2(jx1​)−sin2(jx2​)
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# 定义目标函数defobjective_function(x):
    j =2return3- np.sin(j * x[0])**2- np.sin(j * x[1])**2# 3 - sin(2x1)^2 - sin(2x2)^2# 定义选择函数defcrossover(parents_1, parents_2, crossover_rate):
    num_parents =len(parents_1)# 父母的数量 
    num_crossover =int(crossover_rate * num_parents)# 选择进行交叉的父母对的数量# 选择进行交叉的父母对
    crossover_indices = np.random.choice(num_parents, size=num_crossover, replace=False)# 选择进行交叉的父母对的索引# 复制父母
    copy_parents_1 = np.copy(parents_1)
    copy_parents_2 = np.copy(parents_2)# 进行交叉操作for i in crossover_indices:
        parents_1[i][1]= copy_parents_2[i][1]# 交叉变量x2
        parents_2[i][1]= copy_parents_1[i][1]# 交叉变量x2return parents_1, parents_2

# 定义变异函数defmutate(parents_1, parents_2, mutation_rate):
    num_parents =len(parents_1)# 父母的数量
    num_mutations =int(mutation_rate * num_parents)# 选择进行变异的父母对的数量# 选择进行变异的父母对
    mutation_indices = np.random.choice(num_parents, size=num_mutations, replace=False)# 选择进行变异的父母对的索引# 进行变异操作for i in mutation_indices:
        parents_1[i][1]= np.random.uniform(0,6)# 变异变量x2
        parents_2[i][1]= np.random.uniform(0,6)# 变异变量x2return parents_1, parents_2

# 定义进化算法defevolutionary_algorithm(population_size, num_generations, mutation_rate, crossover_rate):
    bounds =[(0,6),(0,6)]# 变量的取值范围# 保存每个迭代步骤的信息
    evolution_data =[]# 初始化种群
    population = np.random.uniform(bounds[0][0], bounds[0][1], size=(population_size,2))# 设置初始的 best_solution
    best_solution = population[0]# 选择种群中的第一个个体作为初始值
    best_fitness = objective_function(best_solution)# 计算初始值的适应度值for generation inrange(num_generations):# 计算适应度
        fitness_values = np.apply_along_axis(objective_function,1, population)# 找到当前最佳解
        current_best_index = np.argmin(fitness_values)
        current_best_solution = population[current_best_index]
        current_best_fitness = fitness_values[current_best_index]# 绘制每次迭代的三维分布图
        fig = plt.figure()# 创建一个新的图形
        ax = fig.add_subplot(111, projection='3d')# 创建一个三维的坐标系
        ax.scatter(population[:,0], population[:,1], fitness_values, color='black', marker='.', label='Population')# 绘制种群的分布图
        ax.scatter(best_solution[0], best_solution[1], best_fitness, s=100, color='red', marker='o', label='Best Solution')# 绘制最佳解的分布图# 设置坐标轴的标签
        ax.set_xlabel('X1') 
        ax.set_ylabel('X2')
        ax.set_zlabel('f(x)')
        ax.set_title(f'Generation {generation} - Best Fitness: {best_fitness:.6f}')
        ax.legend()# 显示图例
        plt.show()# 显示图形# 更新全局最佳解if current_best_fitness < best_fitness:# 如果当前的最佳解的适应度值小于全局最佳解的适应度值
            best_solution = current_best_solution
            best_fitness = current_best_fitness

        # 保存当前迭代步骤的信息
        evolution_data.append({'generation': generation,'best_solution': best_solution,'best_fitness': best_fitness
        })# 根据适应度值确定复制的数量并且形成繁殖池
        reproduction_ratios = fitness_values / np.sum(fitness_values)# 计算每个个体的适应度值占总适应度值的比例
        sorted_index_ratios = np.argsort(reproduction_ratios)# 对比例进行排序
        half_length =len(sorted_index_ratios)//2# 选择前一半的个体
        first_half_index = sorted_index_ratios[:half_length]# 选择前一半的个体的索引
        new_half_population = population[first_half_index]# 选择前一半的个体
        breeding_pool = np.concatenate((new_half_population, new_half_population))# 将前一半的个体复制一份,形成繁殖池# 选择父母        
        parents_1 = breeding_pool[:half_length]
        parents_2 = breeding_pool[half_length:]# 先获取最后一半的父母
        parents_2 = np.flip(parents_2, axis=0)# 再将父母的顺序反转# 选择和交叉
        parents_1, parents_2 = crossover(parents_1, parents_2, crossover_rate)# 变异
        parents_1, parents_2 = mutate(parents_1, parents_2, mutation_rate)# 更新种群
        population = np.vstack([parents_1, parents_2])return best_solution, best_fitness, evolution_data

# 设置算法参数
population_size =10000
num_generations =40
mutation_rate =0.1# 变异率
crossover_rate =0.4# 交叉率# 运行进化算法
best_solution, best_fitness, evolution_data = evolutionary_algorithm(population_size, num_generations, mutation_rate, crossover_rate)# 输出结果print("最小值:", best_fitness)print("最优解:", best_solution)# 输出每个迭代步骤的最佳适应度值print("每个迭代步骤的最佳适应度值:")for step in evolution_data:print(f"Generation {step['generation']}: {step['best_fitness']}")# 可视化函数曲面和最优解
x1_vals = np.linspace(0,6,100)
x2_vals = np.linspace(0,6,100)
X1, X2 = np.meshgrid(x1_vals, x2_vals)
Z =3- np.sin(2* X1)**2- np.sin(2* X2)**2

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X1, X2, Z, alpha=0.5, cmap='viridis')
ax.scatter(best_solution[0], best_solution[1], best_fitness, color='red', marker='o', label='Best Solution')
ax.set_xlabel('X1')
ax.set_ylabel('X2')
ax.set_zlabel('f(x)')
ax.set_title('Objective Function and Best Solution')
ax.legend()# 绘制适应度值的变化曲线
evolution_df = pd.DataFrame(evolution_data)
plt.figure()
plt.plot(evolution_df['generation'], evolution_df['best_fitness'], label='Best Fitness')
plt.xlabel('Generation')
plt.ylabel('Fitness')
plt.title('Evolution of Fitness')
plt.legend()

plt.show()

4.3 Shubert 函数的最小值

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# 定义目标函数defobjective_function(x):
    result =1for i inrange(1,3):
        inner_sum =0for j inrange(1,6):
            inner_sum += j * np.cos((j +1)* x[i -1]+ j)
        result *= inner_sum
    return result 

# 定义选择函数defcrossover(parents_1, parents_2, crossover_rate):
    num_parents =len(parents_1)# 父母的数量 
    num_crossover =int(crossover_rate * num_parents)# 选择进行交叉的父母对的数量# 选择进行交叉的父母对
    crossover_indices = np.random.choice(num_parents, size=num_crossover, replace=False)# 选择进行交叉的父母对的索引# 复制父母
    copy_parents_1 = np.copy(parents_1)
    copy_parents_2 = np.copy(parents_2)# 进行交叉操作for i in crossover_indices:
        parents_1[i][1]= copy_parents_2[i][1]# 交叉变量x2
        parents_2[i][1]= copy_parents_1[i][1]# 交叉变量x2return parents_1, parents_2

# 定义变异函数defmutate(parents_1, parents_2, mutation_rate):
    num_parents =len(parents_1)# 父母的数量
    num_mutations =int(mutation_rate * num_parents)# 选择进行变异的父母对的数量# 选择进行变异的父母对
    mutation_indices = np.random.choice(num_parents, size=num_mutations, replace=False)# 选择进行变异的父母对的索引# 进行变异操作for i in mutation_indices:
        parents_1[i][1]= np.random.uniform(-10,10)# 变异变量x2
        parents_2[i][1]= np.random.uniform(-10,10)# 变异变量x2return parents_1, parents_2

# 定义进化算法defevolutionary_algorithm(population_size, num_generations, mutation_rate, crossover_rate):
    bounds =[(-10,10),(-10,10)]# 变量的取值范围# 保存每个迭代步骤的信息
    evolution_data =[]# 初始化种群
    population = np.random.uniform(bounds[0][0], bounds[0][1], size=(population_size,2))# 设置初始的 best_solution
    best_solution = population[0]# 选择种群中的第一个个体作为初始值
    best_fitness = objective_function(best_solution)# 计算初始值的适应度值for generation inrange(num_generations):# 计算适应度
        fitness_values = np.apply_along_axis(objective_function,1, population)# 找到当前最佳解
        current_best_index = np.argmin(fitness_values)
        current_best_solution = population[current_best_index]
        current_best_fitness = fitness_values[current_best_index]# 绘制每次迭代的三维分布图
        fig = plt.figure()# 创建一个新的图形
        ax = fig.add_subplot(111, projection='3d')# 创建一个三维的坐标系
        ax.scatter(population[:,0], population[:,1], fitness_values, color='black', marker='.', label='Population')# 绘制种群的分布图
        ax.scatter(current_best_solution[0], current_best_solution[1], current_best_fitness, s=100, color='red', marker='o', label='Best Solution')# 绘制最佳解的分布图# 设置坐标轴的标签
        ax.set_xlabel('X1') 
        ax.set_ylabel('X2')
        ax.set_zlabel('f(x)')
        ax.set_title(f'Generation {generation} - Best Fitness: {current_best_fitness:.6f}')
        ax.legend()# 显示图例
        plt.show()# 显示图形# 更新全局最佳解if current_best_fitness < best_fitness:# 如果当前的最佳解的适应度值小于全局最佳解的适应度值
            best_solution = current_best_solution
            best_fitness = current_best_fitness

        # 保存当前迭代步骤的信息
        evolution_data.append({'generation': generation,'best_solution': best_solution,'best_fitness': best_fitness
        })# 根据适应度值确定复制的数量并且形成繁殖池
        reproduction_ratios = fitness_values / np.sum(fitness_values)# 计算每个个体的适应度值占总适应度值的比例
        sorted_index_ratios = np.argsort(reproduction_ratios)# 对比例进行排序
        half_length =len(sorted_index_ratios)//2# 选择后一半的个体
        first_half_index = sorted_index_ratios[half_length:]# 选择后一半的个体的索引
        new_half_population = population[first_half_index]# 选择后一半的个体
        breeding_pool = np.concatenate((new_half_population, new_half_population))# 将后一半的个体复制一份,形成繁殖池# 选择父母        
        parents_1 = breeding_pool[:half_length]
        parents_2 = breeding_pool[half_length:]# 先获取最后一半的父母
        parents_2 = np.flip(parents_2, axis=0)# 再将父母的顺序反转# 选择和交叉
        parents_1, parents_2 = crossover(parents_1, parents_2, crossover_rate)# 变异
        parents_1, parents_2 = mutate(parents_1, parents_2, mutation_rate)# 更新种群
        population = np.vstack([parents_1, parents_2])return best_solution, best_fitness, evolution_data

# 设置算法参数
population_size =15000
num_generations =40
mutation_rate =0.08# 变异率
crossover_rate =0.2# 交叉率# 运行进化算法
best_solution, best_fitness, evolution_data = evolutionary_algorithm(population_size, num_generations, mutation_rate, crossover_rate)# 输出结果print("最小值:", best_fitness)print("最优解:", best_solution)# 输出每个迭代步骤的最佳适应度值print("每个迭代步骤的最佳适应度值:")for step in evolution_data:print(f"Generation {step['generation']}: {step['best_fitness']}")# 可视化函数曲面和最优解
x1_vals = np.linspace(-10,10,100)
x2_vals = np.linspace(-10,10,100)
X1, X2 = np.meshgrid(x1_vals, x2_vals)
Z = np.zeros_like(X1)for i inrange(Z.shape[0]):for j inrange(Z.shape[1]):
        Z[i, j]= objective_function([X1[i, j], X2[i, j]])

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X1, X2, Z, alpha=0.5, cmap='viridis')
ax.scatter(best_solution[0], best_solution[1], best_fitness, color='red', marker='o', label='Best Solution')
ax.set_xlabel('X1')
ax.set_ylabel('X2')
ax.set_zlabel('f(x)')
ax.set_title('Objective Function and Best Solution')
ax.legend()# 绘制适应度值的变化曲线
evolution_df = pd.DataFrame(evolution_data)
plt.figure()
plt.plot(evolution_df['generation'], evolution_df['best_fitness'], label='Best Fitness')
plt.xlabel('Generation')
plt.ylabel('Fitness')
plt.title('Evolution of Fitness')
plt.legend()

plt.show()

5 附录

5.1 In[1] 输出

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最小值: 1.0000002473000187

最优解: [0.78562713 0.7854951 ]

每个迭代步骤的最佳适应度值:

Generation 0: 1.0000153042180673

Generation 1: 1.0000153042180673

Generation 2: 1.0000153042180673

Generation 3: 1.0000136942409763

Generation 4: 1.0000136942409763

Generation 5: 1.0000136942409763

Generation 6: 1.0000136942409763

Generation 7: 1.0000100419077742

Generation 8: 1.000005565304546

Generation 9: 1.000002458099502

Generation 10: 1.0000022366988228

Generation 11: 1.0000007727585987

Generation 12: 1.0000007727585987

Generation 13: 1.0000007091648468

Generation 14: 1.0000007091648468

Generation 15: 1.0000004471760704

Generation 16: 1.0000004471760704

Generation 17: 1.0000004471760704

Generation 18: 1.0000004471760704

Generation 19: 1.0000002609708571

Generation 20: 1.0000002609708571

Generation 21: 1.0000002609708571

Generation 22: 1.0000002609708571

Generation 23: 1.0000002609708571

Generation 24: 1.0000002609708571

Generation 25: 1.0000002609708571

Generation 26: 1.0000002609708571

Generation 27: 1.0000002609708571

Generation 28: 1.0000002609708571

Generation 29: 1.0000002473000187

Generation 30: 1.0000002473000187

Generation 31: 1.0000002473000187

Generation 32: 1.0000002473000187

Generation 33: 1.0000002473000187

Generation 34: 1.0000002473000187

Generation 35: 1.0000002473000187

Generation 36: 1.0000002473000187

Generation 37: 1.0000002473000187

Generation 38: 1.0000002473000187

Generation 39: 1.0000002473000187

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5.2 In[2] 输出

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最小值: -186.73042323192567

最优解: [-7.70876845 -7.08354764]

每个迭代步骤的最佳适应度值:

Generation 0: -186.59098010602338

Generation 1: -186.59098010602338

Generation 2: -186.59098010602338

Generation 3: -186.59098010602338

Generation 4: -186.70224634663253

Generation 5: -186.70224634663253

Generation 6: -186.70224634663253

Generation 7: -186.70224634663253

Generation 8: -186.70224634663253

Generation 9: -186.70224634663253

Generation 10: -186.70224634663253

Generation 11: -186.71507272172664

Generation 12: -186.71507272172664

Generation 13: -186.7289048406221

Generation 14: -186.73006643615773

Generation 15: -186.73006643615773

Generation 16: -186.73006643615773

Generation 17: -186.73006643615773

Generation 18: -186.73009038074477

Generation 19: -186.73009038074477

Generation 20: -186.73009038074477

Generation 21: -186.73009038074477

Generation 22: -186.73009038074477

Generation 23: -186.73042323192567

Generation 24: -186.73042323192567

Generation 25: -186.73042323192567

Generation 26: -186.73042323192567

Generation 27: -186.73042323192567

Generation 28: -186.73042323192567

Generation 29: -186.73042323192567

Generation 30: -186.73042323192567

Generation 31: -186.73042323192567

Generation 32: -186.73042323192567

Generation 33: -186.73042323192567

Generation 34: -186.73042323192567

Generation 35: -186.73042323192567

Generation 36: -186.73042323192567

Generation 37: -186.73042323192567

Generation 38: -186.73042323192567

Generation 39: -186.73042323192567

output_1_41

output_1_42


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