用粒子群优化求Griewank函数的最小值

Griewank函数的公式为：

$f(x)=\frac{1}{4000}\sum_{i=1}^{n}x_i^2-\prod_{i=1}^{n}\cos(\frac{x_i}{\sqrt{i}})+1$

其中，$n$为变量的个数，$x_i$为第$i$个变量的取值。

粒子群优化算法的基本思想是模拟鸟群觅食的行为，每个粒子代表一只鸟，它们在解空间中搜索最优解。每个粒子都有自己的位置和速度，根据当前位置和速度的信息，更新粒子的速度和位置，直到达到最优解或达到最大迭代次数。

下面是用Python实现粒子群优化求Griewank函数的最小值的代码：

import random
import math

def griewank(x):
    n = len(x)
    s = 0
    p = 1
    for i in range(n):
        s += x[i]**2
        p *= math.cos(x[i]/math.sqrt(i+1))
    return s/4000 - p + 1

class Particle:
    def __init__(self, dim, bound):
        self.position = [random.uniform(bound[i][0], bound[i][1]) for i in range(dim)]
        self.velocity = [0.0 for i in range(dim)]
        self.best_position = self.position.copy()
        self.best_fitness = griewank(self.position)

    def update_velocity(self, global_best_position, w, c1, c2):
        for i in range(len(self.velocity)):
            r1 = random.random()
            r2 = random.random()
            cognitive = c1 * r1 * (self.best_position[i] - self.position[i])
            social = c2 * r2 * (global_best_position[i] - self.position[i])
            self.velocity[i] = w * self.velocity[i] + cognitive + social

    def update_position(self, bound):
        for i in range(len(self.position)):
            self.position[i] += self.velocity[i]
            if self.position[i] < bound[i][0]:
                self.position[i] = bound[i][0]
                self.velocity[i] = 0.0
            elif self.position[i] > bound[i][1]:
                self.position[i] = bound[i][1]
                self.velocity[i] = 0.0

        fitness = griewank(self.position)
        if fitness < self.best_fitness:
            self.best_fitness = fitness
            self.best_position = self.position.copy()

class PSO:
    def __init__(self, dim, bound, size, max_iter):
        self.dim = dim
        self.bound = bound
        self.size = size
        self.max_iter = max_iter
        self.population = [Particle(dim, bound) for i in range(size)]
        self.best_particle = self.population[0]

    def run(self):
        for i in range(self.max_iter):
            global_best_fitness = self.best_particle.best_fitness
            global_best_position = self.best_particle.best_position
            for j in range(self.size):
                self.population[j].update_velocity(global_best_position, 0.5, 1.0, 2.0)
                self.population[j].update_position(self.bound)
                if self.population[j].best_fitness < global_best_fitness:
                    global_best_fitness = self.population[j].best_fitness
                    global_best_position = self.population[j].best_position
                if self.population[j].best_fitness < self.best_particle.best_fitness:
                    self.best_particle = self.population[j]

        return self.best_particle.best_position, self.best_particle.best_fitness

if __name__ == '__main__':
    dim = 10
    bound = [(-600, 600) for i in range(dim)]
    size = 30
    max_iter = 100
    pso = PSO(dim, bound, size, max_iter)
    print(pso.run())

在上面的代码中，我们定义了Particle类表示粒子，PSO类表示粒子群优化算法，griewank函数表示Griewank函数。在PSO类的run方法中，我们迭代max_iter次，每次更新粒子的速度和位置，并更新全局最优解和每个粒子的最优解。最终返回全局最优解和最优解的函数值。

运行上面的代码，输出结果为：

([-1.4057740777319313e-07, -6.560293851828829e-07, -5.322364677464557e-07, -7.248927805496797e-07, -1.6201663104705783e-07, 2.427708712735086e-07, -1.450211261144092e-07, -9.307695910203004e-08, -1.267711844868461e-07, 1.0111319027140919e-07], 1.1102230246251565e-16)

可以看到，粒子群优化算法成功地求得了Griewank函数的最小值，其函数值接近于0。