5大智能算法优化标准测试函数对比（Python实现）

标准测试函数对比（Python实现）

一、问题背景

智能优化算法是解决复杂优化问题的重要工具。本文选取5种经典智能算法，在5个标准测试函数上进行对比实验，全面评估各算法的性能。

5种智能算法

算法	缩写	灵感来源	提出时间
遗传算法	GA	自然选择与遗传	1975年
粒子群算法	PSO	鸟群觅食行为	1995年
模拟退火算法	SA	金属退火过程	1983年
差分进化算法	DE	种群进化	1995年
蚁群算法	ACO	蚂蚁觅食行为	1992年

5个标准测试函数

函数	公式	搜索范围
Sphere	f(x)=∑i=1nxi2f(x)=\sum_{i=1}^n x_i^2f(x)=∑i=1nxi2	$-5,5$
Rosenbrock	f(x)=∑i=1n−1 $100(xi+1-xi2)2+(xi-1)2$ f(x)=\sum_{i=1}^{n-1} $100(x_{i+1}-x_i\^2)\^2+(x_i-1)\^2$ f(x)=∑i=1n−1 $100(xi+1-xi2)2+(xi-1)2$	$-2,2$
Rastrigin	f(x)=10n+∑i=1n $xi2-10cos(2πxi)$ f(x)=10n+\sum_{i=1}^n $x_i\^2-10\\cos(2\\pi x_i)$ f(x)=10n+∑i=1n $xi2-10cos(2πxi)$	$-5,5$
Ackley	f(x)=−20exp⁡(−0.21n∑xi2)−exp⁡(1n∑cos⁡(2πxi))+20+ef(x)=-20\exp(-0.2\sqrt{\frac{1}{n}\sum x_i^2})-\exp(\frac{1}{n}\sum\cos(2\pi x_i))+20+ef(x)=−20exp(−0.2n1∑xi2 )−exp(n1∑cos(2πxi))+20+e	$-5,5$
Griewank	f(x)=14000∑xi2−∏cos⁡(xii)+1f(x)=\frac{1}{4000}\sum x_i^2-\prod\cos(\frac{x_i}{\sqrt{i}})+1f(x)=40001∑xi2−∏cos(i xi)+1	$-5,5$

二、Python实现

2.1 测试函数定义

python 复制代码

import numpy as np

def sphere(x):
    return np.sum(x ** 2)

def rosenbrock(x):
    return np.sum(100 * (x[1:] - x[:-1]**2)**2 + (x[:-1] - 1)**2)

def rastrigin(x):
    n = len(x)
    return 10 * n + np.sum(x**2 - 10 * np.cos(2 * np.pi * x))

def ackley(x):
    n = len(x)
    sum1, sum2 = np.sum(x**2), np.sum(np.cos(2 * np.pi * x))
    return -20 * np.exp(-0.2 * np.sqrt(sum1 / n)) - np.exp(sum2 / n) + 20 + np.e

def griewank(x):
    n = len(x)
    return np.sum(x**2) / 4000 - np.prod(np.cos(x / np.sqrt(np.arange(1, n+1)))) + 1

2.2 遗传算法（GA）

python 复制代码

def ga(func, dim=10, bounds=(-5,5), pop_size=50, max_iter=100):
    pop = np.random.uniform(bounds[0], bounds[1], (pop_size, dim))
    best = float('inf')
    best_history = []
    for gen in range(max_iter):
        fitness = np.array([func(p) for p in pop])
        if np.min(fitness) < best:
            best = np.min(fitness)
        best_history.append(best)
        new_pop = np.zeros_like(pop)
        for i in range(pop_size):
            cand = np.random.choice(pop_size, 3, replace=False)
            p1 = pop[cand[np.argmin(fitness[cand])]]
            cand = np.random.choice(pop_size, 3, replace=False)
            p2 = pop[cand[np.argmin(fitness[cand])]]
            mask = np.random.rand(dim) < 0.8
            child = np.where(mask, p1, p2)
            mut = np.random.rand(dim) < 0.1
            child[mut] += np.random.normal(0, 0.5, np.sum(mut))
            child = np.clip(child, bounds[0], bounds[1])
            new_pop[i] = child
        pop = new_pop
    return best_history

2.3 粒子群算法（PSO）

python 复制代码

def pso(func, dim=10, bounds=(-5,5), swarm=50, max_iter=100):
    pos = np.random.uniform(bounds[0], bounds[1], (swarm, dim))
    vel = np.random.uniform(-1, 1, (swarm, dim))
    pbest, pbest_val = pos.copy(), np.array([func(p) for p in pos])
    gbest_idx = np.argmin(pbest_val)
    gbest, gbest_val = pos[gbest_idx].copy(), pbest_val[gbest_idx]
    best_history = [gbest_val]
    w, c1, c2 = 0.7, 1.5, 1.5
    for t in range(max_iter):
        r1, r2 = np.random.rand(swarm, dim), np.random.rand(swarm, dim)
        vel = w * vel + c1 * r1 * (pbest - pos) + c2 * r2 * (gbest - pos)
        pos = np.clip(pos + vel, bounds[0], bounds[1])
        val = np.array([func(p) for p in pos])
        improved = val < pbest_val
        pbest[improved], pbest_val[improved] = pos[improved], val[improved]
        if np.min(val) < gbest_val:
            gbest_idx = np.argmin(val)
            gbest, gbest_val = pos[gbest_idx].copy(), val[gbest_idx]
        best_history.append(gbest_val)
    return best_history

2.4 模拟退火算法（SA）

python 复制代码

def sa(func, dim=10, bounds=(-5,5), max_iter=100):
    x = np.random.uniform(bounds[0], bounds[1], dim)
    fx, best = func(x), func(x)
    best_history = [best]
    T = 100
    for t in range(max_iter):
        for _ in range(10):
            x_new = np.clip(x + np.random.normal(0, 1, dim), bounds[0], bounds[1])
            f_new = func(x_new)
            if f_new < fx or np.random.rand() < np.exp(-(f_new - fx) / T):
                x, fx = x_new, f_new
                if fx < best:
                    best = fx
        T = max(T * 0.95, 0.01)
        best_history.append(best)
    return best_history

2.5 差分进化算法（DE）

python 复制代码

def de(func, dim=10, bounds=(-5,5), pop_size=50, max_iter=100):
    pop = np.random.uniform(bounds[0], bounds[1], (pop_size, dim))
    fitness = np.array([func(p) for p in pop])
    best, F, CR = np.min(fitness), 0.5, 0.9
    best_history = [best]
    for t in range(max_iter):
        for i in range(pop_size):
            idxs = [j for j in range(pop_size) if j != i]
            a, b, c = pop[np.random.choice(idxs, 3, replace=False)]
            mutant = np.clip(a + F * (b - c), bounds[0], bounds[1])
            trial = np.where(np.random.rand(dim) < CR, mutant, pop[i])
            ft = func(trial)
            if ft < fitness[i]:
                pop[i], fitness[i] = trial, ft
        if np.min(fitness) < best:
            best = np.min(fitness)
        best_history.append(best)
    return best_history

2.6 蚁群算法（ACO - 连续优化版）

python 复制代码

def aco(func, dim=10, bounds=(-5,5), ants=50, max_iter=100):
    solutions = np.random.uniform(bounds[0], bounds[1], (ants, dim))
    fitness = np.array([func(s) for s in solutions])
    best, tau = np.min(fitness), np.ones((ants, dim))
    best_history = [best]
    for t in range(max_iter):
        for i in range(ants):
            if np.random.rand() < 0.3:
                solutions[i] += np.random.normal(0, 0.1, dim) * tau[i]
                solutions[i] = np.clip(solutions[i], bounds[0], bounds[1])
                f_new = func(solutions[i])
                if f_new < fitness[i]:
                    fitness[i] = f_new
        tau = tau * 0.9 + 1.0 / (fitness.reshape(-1, 1) + 1e-10)
        if np.min(fitness) < best:
            best = np.min(fitness)
        best_history.append(best)
    return best_history

2.7 主程序

python 复制代码

test_funcs = [
    ("Sphere", sphere, (-5, 5)),
    ("Rosenbrock", rosenbrock, (-2, 2)),
    ("Rastrigin", rastrigin, (-5, 5)),
    ("Ackley", ackley, (-5, 5)),
    ("Griewank", griewank, (-5, 5))
]
algorithms = [("GA", ga), ("PSO", pso), ("SA", sa),
              ("DE", de), ("ACO", aco)]

dim, max_iter = 10, 100

for fname, func, bounds in test_funcs:
    print(f"\n===== {fname} =====")
    for aname, algo in algorithms:
        history = algo(func, dim=dim, bounds=bounds, max_iter=max_iter)
        print(f"  {aname}: 最优={history[-1]:.6f}")

三、运行结果

3.1 控制台输出

复制代码

===== Sphere =====
  GA: 最优=0.008459
  PSO: 最优=0.000010
  SA: 最优=4.292638
  DE: 最优=0.000085
  ACO: 最优=38.025978

===== Rosenbrock =====
  GA: 最优=8.530541
  PSO: 最优=7.509120
  SA: 最优=27.989428
  DE: 最优=6.385049
  ACO: 最优=586.418290

===== Rastrigin =====
  GA: 最优=7.231806
  PSO: 最优=12.320289
  SA: 最优=59.652024
  DE: 最优=41.829613
  ACO: 最优=87.136393

===== Ackley =====
  GA: 最优=0.127063
  PSO: 最优=0.001506
  SA: 最优=6.422770
  DE: 最优=0.034487
  ACO: 最优=7.255058

===== Griewank =====
  GA: 最优=0.008372
  PSO: 最优=0.007398
  SA: 最优=0.838557
  DE: 最优=0.077606
  ACO: 最优=0.853547

3.2 结果对比

函数	GA	PSO	SA	DE	ACO
Sphere	8.46e-3	1.00e-5	4.29	8.50e-5	38.03
Rosenbrock	8.53	6.39(DE)	27.99	6.39	586.42
Rastrigin	7.23	12.32	59.65	41.83	87.14
Ackley	0.127	0.002	6.42	0.035	7.26
Griewank	0.008	0.007	0.84	0.078	0.85

3.3 收敛曲线对比

图1：五种算法在五个测试函数上的收敛曲线对比

3.4 各函数单独收敛曲线

图2：Sphere函数收敛对比

图3：Rosenbrock函数收敛对比

图4：Rastrigin函数收敛对比

图5：Ackley函数收敛对比

图6：Griewank函数收敛对比

四、结果分析

PSO综合表现最佳：在Sphere、Ackley、Griewank上均取得最优结果，收敛速度快
DE在Rosenbrock上最优：Rosenbrock是多峰函数，DE的差分变异策略效果显著
GA在Rastrigin上最优：GA的交叉变异机制适合Rastrigin的多峰搜索
SA收敛较慢：模拟退火在有限迭代内难以充分收敛
ACO（连续优化版）在简单函数上表现一般：标准ACO更适合离散组合优化

五、参数设置

算法	参数	数值
GA	种群/交叉率/变异率	50 / 0.8 / 0.1
PSO	种群/w/c1/c2	50 / 0.7 / 1.5 / 1.5
SA	初温/终温/降温系数	100 / 0.01 / 0.95
DE	种群/F/CR	50 / 0.5 / 0.9
ACO	蚂蚁数/挥发系数	50 / 0.1

六、总结

本文使用Python实现了5种经典智能优化算法（GA、PSO、SA、DE、ACO），并在5个标准测试函数（Sphere、Rosenbrock、Rastrigin、Ackley、Griewank）上进行了对比实验。实验结果表明：

PSO在大多数函数上表现最优
DE在复杂多峰函数上具有优势
GA在特定问题上表现突出
不同算法各有适用场景

参考资料

Kennedy J, Eberhart R. Particle swarm optimization. 1995.
Storn R, Price K. Differential evolution-a simple and efficient heuristic. 1997.
Dorigo M, Stutzle T. Ant Colony Optimization. 2004.

完整代码可直接运行，如有问题欢迎交流！