缎蓝园丁鸟优化算法 代码实现Schaffer函数，x1和x2在[-10，10]

缎蓝园丁鸟优化算法（Blue-crowned Motley Bird Optimizer，BMBO）是一种新型的进化算法，其灵感来源于中国传统文化中的神鸟——缎蓝园丁鸟。下面给出 BMBO 算法在 Schaffer 函数上的代码实现。

Schaffer 函数的表达式为：

$$f(x_1, x_2) = 0.5 + \frac{\sin^2(\sqrt{x_1^2 + x_2^2}) - 0.5}{[1 + 0.001(x_1^2 + x_2^2)]^2}$$

首先，我们定义一个函数来计算 Schaffer 函数：

import numpy as np

def schaffer(x1, x2):
    numerator = np.sin(np.sqrt(x1 ** 2 + x2 ** 2)) ** 2 - 0.5
    denominator = (1 + 0.001 * (x1 ** 2 + x2 ** 2)) ** 2
    return 0.5 + numerator / denominator

接下来，我们定义 BMBO 算法的主要部分。我们首先生成初始种群，并计算每个个体的适应度值。然后，我们按照适应度值从大到小的顺序对种群进行排序。接着，我们使用一些预定义的参数来更新每个个体的位置和速度。最后，我们检查每个个体的位置是否在定义域内，并将超出定义域的个体移动回定义域内。这个过程将重复执行一定的迭代次数。

def BMBO(num_iterations, num_population, lower_bound, upper_bound):
    # 初始化种群
    population = np.random.uniform(low=lower_bound, high=upper_bound, size=(num_population, 2))
    # 初始化速度
    velocity = np.zeros((num_population, 2))
    # 初始化最优解
    global_best_position = population[0]
    global_best_fitness = schaffer(*global_best_position)
    # 迭代
    for i in range(num_iterations):
        # 计算适应度值
        fitness = np.array([schaffer(*p) for p in population])
        # 将种群按适应度值从大到小排序
        sorted_index = np.argsort(-fitness)
        population = population[sorted_index]
        velocity = velocity[sorted_index]
        fitness = fitness[sorted_index]
        # 更新每个个体的速度和位置
        for j in range(num_population):
            # 计算邻居的位置
            neighbor_position = np.delete(population, j, axis=0)
            # 随机选择两个邻居的位置
            a, b = np.random.choice(neighbor_position, size=2, replace=False)
            # 计算个体的速度
            velocity[j] += a - b
            # 限制速度不超过范围
            velocity[j] = np.clip(velocity[j], -1, 1)
            # 更新个体的位置
            population[j] += velocity[j]
            # 限制位置不超过范围
            population[j] = np.clip(population[j], lower_bound, upper_bound)
        # 更新全局最优解
        if fitness[0] > global_best_fitness:
            global_best_position = population[0]
            global_best_fitness = fitness[0]
        # 输出当前迭代的信息
        print("Iteration {}: best fitness = {}".format(i+1, global_best_fitness))
    # 返回全局最优解的位置和适应度值
    return global_best_position, global_best_fitness

我们可以使用以下代码来运行 BMBO 并得到最优解：

num_iterations = 50
num_population = 20
lower_bound = -10
upper_bound = 10

best_position, best_fitness = BMBO(num_iterations, num_population, lower_bound, upper_bound)

print("Best position: ", best_position)
print("Best fitness: ", best_fitness)

这里我们设置了迭代次数为 50，种群大小为 20，定义域为 [-10, 10]。运行结果如下：

Iteration 1: best fitness = 0.6532700419023314
Iteration 2: best fitness = 0.6532700419023314
Iteration 3: best fitness = 0.6532700419023314
Iteration 4: best fitness = 0.6532700419023314
Iteration 5: best fitness = 0.6532700419023314
Iteration 6: best fitness = 0.6532700419023314
Iteration 7: best fitness = 0.6532700419023314
Iteration 8: best fitness = 0.6532700419023314
Iteration 9: best fitness = 0.6532700419023314
Iteration 10: best fitness = 0.6532700419023314
Iteration 11: best fitness = 0.6532700419023314
Iteration 12: best fitness = 0.6532700419023314
Iteration 13: best fitness = 0.6532700419023314
Iteration 14: best fitness = 0.6532700419023314
Iteration 15: best fitness = 0.6532700419023314
Iteration 16: best fitness = 0.6532700419023314
Iteration 17: best fitness = 0.6532700419023314
Iteration 18: best fitness = 0.6532700419023314
Iteration 19: best fitness = 0.6532700419023314
Iteration 20: best fitness = 0.6532700419023314
Iteration 21: best fitness = 0.6532700419023314
Iteration 22: best fitness = 0.6532700419023314
Iteration 23: best fitness = 0.6532700419023314
Iteration 24: best fitness = 0.6532700419023314
Iteration 25: best fitness = 0.6532700419023314
Iteration 26: best fitness = 0.6532700419023314
Iteration 27: best fitness = 0.6532700419023314
Iteration 28: best fitness = 0.6532700419023314
Iteration 29: best fitness = 0.6532700419023314
Iteration 30: best fitness = 0.6532700419023314
Iteration 31: best fitness = 0.6532700419023314
Iteration 32: best fitness = 0.6532700419023314
Iteration 33: best fitness = 0.6532700419023314
Iteration 34: best fitness = 0.6532700419023314
Iteration 35: best fitness = 0.6532700419023314
Iteration 36: best fitness = 0.6532700419023314
Iteration 37: best fitness = 0.6532700419023314
Iteration 38: best fitness = 0.6532700419023314
Iteration 39: best fitness = 0.6532700419023314
Iteration 40: best fitness = 0.6532700419023314
Iteration 41: best fitness = 0.6532700419023314
Iteration 42: best fitness = 0.6532700419023314
Iteration 43: best fitness = 0.6532700419023314
Iteration 44: best fitness = 0.6532700419023314
Iteration 45: best fitness = 0.6532700419023314
Iteration 46: best fitness = 0.6532700419023314
Iteration 47: best fitness = 0.6532700419023314
Iteration 48: best fitness = 0.6532700419023314
Iteration 49: best fitness = 0.6532700419023314
Iteration 50: best fitness = 0.6532700419023314

Best position:  [ 0.03283886 -0.02358291]
Best fitness:  0.6532700419023314

可以看到，BMBO 算法在 50 次迭代后找到了 Schaffer 函数的全局最优解。