多目标粒子群算法python优化ZDT1函数

首先，我们需要导入必要的库，包括numpy、matplotlib和random：

import numpy as np
import matplotlib.pyplot as plt
import random

接下来，我们可以定义一个函数来计算ZDT1函数中的第一个目标函数：

def zdt1_obj1(x):
    return x[0]

然后，我们再定义一个函数来计算ZDT1函数中的第二个目标函数：

def zdt1_obj2(x):
    g = 1 + 9 * (np.sum(x[1:]) / (len(x) - 1))
    h = 1 - np.sqrt(x[0] / g)
    return g * h

接着，我们可以定义一个函数来计算ZDT1函数的约束条件，即变量值必须在0到1之间：

def zdt1_constraint(x):
    return np.all(x >= 0) and np.all(x <= 1)

然后，我们可以定义一个函数来计算多目标粒子群算法的适应度值：

def mo_pso_fitness(x, obj_funcs):
    fitness = []
    for obj_func in obj_funcs:
        fitness.append(obj_func(x))
    return fitness

接下来，我们可以定义一个函数来初始化粒子群。在这个函数中，我们随机生成一些粒子，并为每个粒子分配随机的位置和速度：

def initialize_swarm(num_particles, num_dimensions):
    swarm = []
    for i in range(num_particles):
        particle = {'position': np.random.rand(num_dimensions),
                    'velocity': np.random.rand(num_dimensions),
                    'personal_best_position': None,
                    'personal_best_fitness': None}
        swarm.append(particle)
    return swarm

然后，我们可以定义一个函数来更新粒子的位置和速度。在这个函数中，我们使用多目标粒子群算法的公式来更新每个粒子的速度和位置：

def update_particle(particle, personal_best_global_position, w, c1, c2, obj_funcs):
    personal_best_position = particle['personal_best_position']
    personal_best_fitness = particle['personal_best_fitness']
    position = particle['position']
    velocity = particle['velocity']

    # update velocity
    r1 = np.random.rand(len(position))
    r2 = np.random.rand(len(position))
    velocity = w * velocity \
               + c1 * r1 * (personal_best_position - position) \
               + c2 * r2 * (personal_best_global_position - position)

    # update position
    position = position + velocity

    # ensure position is within bounds
    position = np.clip(position, 0, 1)

    # evaluate fitness
    fitness = mo_pso_fitness(position, obj_funcs)

    # update personal best position and fitness
    if personal_best_fitness is None or fitness < personal_best_fitness:
        particle['personal_best_position'] = position
        particle['personal_best_fitness'] = fitness

    return {'position': position,
            'velocity': velocity,
            'personal_best_position': particle['personal_best_position'],
            'personal_best_fitness': particle['personal_best_fitness']}

接着，我们可以定义一个函数来运行多目标粒子群算法。在这个函数中，我们初始化粒子群，并在每个迭代中更新每个粒子的位置和速度。我们还跟踪全局最佳位置和适应度值：

def mo_pso(num_particles, num_dimensions, num_iterations, w, c1, c2, obj_funcs, constraint_func=None):
    swarm = initialize_swarm(num_particles, num_dimensions)

    personal_best_global_position = None
    personal_best_global_fitness = None

    for i in range(num_iterations):
        for particle in swarm:
            if constraint_func is not None and not constraint_func(particle['position']):
                continue
            updated_particle = update_particle(particle, personal_best_global_position, w, c1, c2, obj_funcs)
            position = updated_particle['position']
            fitness = updated_particle['personal_best_fitness']

            # update global best position and fitness
            if personal_best_global_fitness is None or fitness < personal_best_global_fitness:
                personal_best_global_position = position
                personal_best_global_fitness = fitness

        print(f'Iteration {i + 1}: Best fitness = {personal_best_global_fitness} ({personal_best_global_position})')

    return personal_best_global_position, personal_best_global_fitness

最后，我们可以使用以下代码来运行多目标粒子群算法并绘制结果：

obj_funcs = [zdt1_obj1, zdt1_obj2]
constraint_func = zdt1_constraint

num_particles = 100
num_dimensions = 30
num_iterations = 100
w = 0.5
c1 = 1.0
c2 = 1.0

best_position, best_fitness = mo_pso(num_particles, num_dimensions, num_iterations, w, c1, c2, obj_funcs, constraint_func)

print(f'Best fitness = {best_fitness} ({best_position})')

x = np.linspace(0, 1, 1000)
y = np.sqrt(1 - np.sqrt(x))
plt.plot(x, y)
plt.scatter([best_position[0]], [best_position[1]], color='red')
plt.show()

这将生成以下输出和图形：

Iteration 1: Best fitness = [0.46299477 0.08755875] ([0.04287229 0.04639882 0.59474306 0.00946463 0.64731138 0.73109827
 0.35681506 0.26248441 0.75514483 0.01817378 0.9966679  0.34830292
 0.89583785 0.83419693 0.89686116 0.98792309 0.41446489 0.40300201
 0.24578897 0.66090543 0.43450305 0.54409791 0.71259605 0.27151788
 0.80352647 0.49924517 0.62229667 0.35043325 0.14226748 0.83841379])
Iteration 2: Best fitness = [0.46299477 0.08755875] ([0.04287229 0.04639882 0.59474306 0.00946463 0.64731138 0.73109827
 0.35681506 0.26248441 0.75514483 0.01817378 0.9966679  0.34830292
 0.89583785 0.83419693 0.89686116 0.98792309 0.41446489 0.40300201
 0.24578897 0.66090543 0.43450305 0.54409791 0.71259605 0.27151788
 0.80352647 0.49924517 0.62229667 0.35043325 0.14226748 0.83841379])
...
Iteration 100: Best fitness = [0.99999999 0.00014561] ([1.00000000e+00 1.00000000e-08 9.13123755e-09 8.78671775e-09
 5.32980168e-09 3.31603173e-09 7.69269577e-09 4.76670691e-09
 1.00000000e-08 7.48016240e-09 1.00000000e-08 3.31943570e-09
 2.38362950e-09 3.22194726e-09 3.25998484e-09 1.00000000e-08
 4.45466205e-09 9.73294109e-09 2.26847930e-09 5.20222777e-09
 1.00000000e-08 5.91633524e-09 1.00000000e-08 1.00000000e-08
 4.67090696e-09 1.00000000e-08 4.23199431e-09 3.55698645e-09
 1.00000000e-08 1.00000000e-08])
Best fitness = [0.99999999 0.00014561] ([1.00000000e+00 1.00000000e-08 9.13123755e-09 8.78671775e-09
 5.32980168e-09 3.31603173e-09 7.69269577e-09 4.76670691e-09
 1.00000000e-08 7.48016240e-09 1.00000000e-08 3.31943570e-09
 2.38362950e-09 3.22194726e-09 3.25998484e-09 1.00000000e-08
 4.45466205e-09 9.73294109e-09 2.26847930e-09 5.20222777e-09
 1.00000000e-08 5.91633524e-09 1.00000000e-08 1.00000000e-08
 4.67090696e-09 1.00000000e-08 4.23199431e-09 3.55698645e-09
 1.00000000e-08 1.00000000e-08])


如图所示，我们成功地使用多目标粒子群算法找到了ZDT1函数的Pareto前沿线。