python多目标优化 nsga-ii code

下面是一个简单的Python多目标优化NSGA-II代码示例：

import random
import math

# 定义目标函数
def objectives(x):
    f1 = x[0]**2 + x[1]**2
    f2 = (x[0]-1)**2 + x[1]**2
    return [f1, f2]

# 定义NSGA-II算法
def nsga2(population, n_obj, n_var, n_pop, n_gen):
    # 初始化种群
    for i in range(n_pop):
        for j in range(n_var):
            population[i][j] = random.uniform(-5, 5)

    # 进行n_gen代的进化
    for g in range(n_gen):
        # 计算每个个体的目标函数值
        for i in range(n_pop):
            population[i][n_var:] = objectives(population[i][:n_var])

        # 对每个目标函数进行排序和计算拥挤度
        fronts = [[] for i in range(n_pop)]
        for i in range(n_pop):
            fronts[i] = []
            for j in range(n_pop):
                if i == j:
                    continue
                if dominates(population[i][n_var:], population[j][n_var:]):
                    fronts[i].append(j)
                elif dominates(population[j][n_var:], population[i][n_var:]):
                    population[i][-1] += 1
            if population[i][-1] == 0:
                fronts[i] = [i]

        # 选择下一代种群
        new_pop = []
        front_index = 0
        while len(new_pop) < n_pop:
            if len(fronts[front_index]) + len(new_pop) <= n_pop:
                new_pop += fronts[front_index]
                front_index += 1
            else:
                crowding_distances = [0] * len(fronts[front_index])
                for i in range(n_obj):
                    sorted_index = sorted(range(len(fronts[front_index])), key=lambda x: population[fronts[front_index][x]][n_var+i])
                    crowding_distances[sorted_index[0]] += math.inf
                    crowding_distances[sorted_index[-1]] += math.inf
                    for j in range(1, len(sorted_index)-1):
                        crowding_distances[sorted_index[j]] += population[fronts[front_index][sorted_index[j+1]]][n_var+i] - population[fronts[front_index][sorted_index[j-1]]][n_var+i]
                sorted_index = sorted(range(len(fronts[front_index])), key=lambda x: crowding_distances[x], reverse=True)
                new_pop += [fronts[front_index][i] for i in sorted_index[:n_pop-len(new_pop)]]

        # 生成新的种群
        population = [population[i] for i in new_pop]

    # 返回最终种群
    return population

# 判断一个解是否被另一个解支配
def dominates(x, y):
    return all([x[i] <= y[i] for i in range(len(x))]) and any([x[i] < y[i] for i in range(len(x))])

# 测试代码
if __name__ == '__main__':
    n_obj = 2
    n_var = 2
    n_pop = 50
    n_gen = 100
    population = [[0]*(n_var+n_obj+1) for i in range(n_pop)]
    pareto_front = nsga2(population, n_obj, n_var, n_pop, n_gen)
    for i in range(n_pop):
        print(pareto_front[i][:n_var], pareto_front[i][n_var:])

该代码实现了一个简单的NSGA-II算法，用于求解具有2个目标函数和2个决策变量的多目标优化问题。在该代码中，目标函数为$f_1(x)=x_1^2+x_2^2$和$f_2(x)=(x_1-1)^2+x_2^2$，决策变量的范围为$x_1\in[-5,5]$和$x_2\in[-5,5]$。