用Python代码实现MOEA/D

MOEA/D (Multi-Objective Evolutionary Algorithm Based on Decomposition)是一种多目标优化算法。它将多目标优化问题转化为多个单目标优化问题，并通过分解技术来解决这些单目标优化问题。下面是用Python实现MOEA/D的简单步骤：

1. 定义目标函数和变量范围。
2. 初始化种群，并为每个个体分配权重向量。
3. 计算每个个体的适应度值，并根据适应度值对种群进行排序。
4. 选择父代，并根据父代生成子代。
5. 对生成的子代进行修剪，保留最优的子代。
6. 更新参考点和权重向量，并将新的参考点和权重向量用于下一代进化。

下面是一个简单的Python代码示例：

import numpy as np

# 定义目标函数
def obj_func(x):
    f1 = x**2 + x**2
    f2 = (x-1)**2 + x**2
    return [f1, f2]

# 定义变量范围
x_min = np.array([-5, -5])
x_max = np.array([5, 5])

# 初始化种群和权重向量
pop_size = 100
num_obj = 2
pop = np.random.uniform(low=x_min, high=x_max, size=(pop_size, len(x_min)))
weights = np.random.uniform(low=0, high=1, size=(pop_size, num_obj))
weights = weights / np.sum(weights, axis=1)[:, None]

# 进行进化迭代
num_generations = 100
for gen in range(num_generations):
    # 计算适应度值并排序
    fitness = np.array([obj_func(x) for x in pop])
    ranks = np.argsort(np.argsort(-fitness.dot(weights.T), axis=0), axis=0)
    # 选择父代并生成子代
    parents = np.random.choice(pop_size, size=pop_size)
    children = np.zeros_like(pop)
    for i in range(pop_size):
        parent1 = parents[i]
        parent2 = parents[(i+1) % pop_size]
        child = pop[parent1] + 0.5*(pop[parent2]-pop[parent1])
        child += 0.01*np.random.normal(size=len(x_min))
        child = np.clip(child, x_min, x_max)
        children[i] = child
    # 修剪子代并更新参考点和权重向量
    fitness_children = np.array([obj_func(x) for x in children])
    ranks_children = np.argsort(np.argsort(-fitness_children.dot(weights.T), axis=0), axis=0)
    pop_new = np.zeros_like(pop)
    for i in range(pop_size):
        if ranks_children[i] <= ranks[i]:
            pop_new[i] = children[i]
        else:
            pop_new[i] = pop[i]
    pop = pop_new
    weights = np.clip(weights + 0.01*np.random.normal(size=(pop_size, num_obj)), 0, 1)
    weights = weights / np.sum(weights, axis=1)[:, None]