sphere函数差分进化算法

Sphere函数是一个常用的基准测试函数，其公式为：

f(x) = x1^2 + x2^2 + ... + xn^2

其中x为n维向量。差分进化算法（Differential Evolution, DE）是一种常用的优化算法，它通过对候选解进行向量差分来产生新的解，并根据一定的策略选择更优的解作为下一代种群的基础。

下面是使用差分进化算法求解Sphere函数的Python代码示例：

import numpy as np

def sphere(x):
    return np.sum(np.square(x))

def de(fobj, bounds, mut=0.8, crossp=0.7, popsize=20, its=1000):
    dimensions = len(bounds)
    pop = np.random.rand(popsize, dimensions)
    min_b, max_b = np.asarray(bounds).T
    diff = np.fabs(min_b - max_b)
    pop_denorm = min_b + pop * diff
    fitness = np.asarray([fobj(ind) for ind in pop_denorm])
    best_idx = np.argmin(fitness)
    best = pop_denorm[best_idx]
    for i in range(its):
        for j in range(popsize):
            idxs = [idx for idx in range(popsize) if idx != j]
            a, b, c = pop[np.random.choice(idxs, 3, replace=False)]
            mutant = np.clip(a + mut * (b - c), 0, 1)
            cross_points = np.random.rand(dimensions) < crossp
            if not np.any(cross_points):
                cross_points[np.random.randint(0, dimensions)] = True
            trial = np.where(cross_points, mutant, pop[j])
            trial_denorm = min_b + trial * diff
            f = fobj(trial_denorm)
            if f < fitness[j]:
                fitness[j] = f
                pop[j] = trial
                if f < fitness[best_idx]:
                    best_idx = j
                    best = trial_denorm
        yield best, fitness[best_idx]

bounds = [(0, 10)] * 10 # 10维变量范围在0到10之间
for solution, fitness in de(sphere, bounds):
    print(solution, fitness)

代码中使用了生成随机种群、向量差分、选择优秀解等经典的差分进化算法操作，通过迭代不断优化种群，最终得到了Sphere函数的最优解。