### 实现23个标准测试函数用于优化算法评估 为了验证不同优化算法的有效性和性能，通常会采用一系列标准化的测试函数。这些测试函数具有不同的特性，可以全面考察优化算法的能力。下面展示如何编写 Python 代码来定义这23个常用的基准测试函数。 #### 定义23个测试函数 ```python import numpy as np def sphere(x): """Sphere function""" return sum(xi ** 2 for xi in x) def rosenbrock(x): """Rosenbrock's valley or banana function""" n = len(x) total = 0 for i in range(n - 1): total += 100 * (x[i + 1] - x[i] ** 2) ** 2 + (1 - x[i]) ** 2 return total def rastrigin(x): """Rastrigin function""" A = 10 d = len(x) return A * d + sum([(xi ** 2 - A * np.cos(2 * np.pi * xi)) for xi in x]) def griewank(x): """Griewank function""" part1 = sum([xi ** 2 for xi in x]) part2 = np.prod([np.cos(xi / np.sqrt(i + 1)) for i, xi in enumerate(x)]) return 1 + (part1 / 4000) - part2 def ackley(x): """Ackley function""" a = 20 b = 0.2 c = 2 * np.pi d = len(x) sum_sq_term = -a * np.exp(-b * np.sqrt(sum(xi ** 2 for xi in x) / d)) cos_term = -np.exp(sum(np.cos(c * xi) for xi in x) / d) return a + np.e + sum_sq_term + cos_term def schwefel(x): """Schwefel function""" dim = len(x) return 418.9829 * dim - sum([xi * np.sin(np.abs(xi) ** 0.5) for xi in x]) def zakharov(x): """Zakharov function""" first_sum = sum([xi ** 2 for xi in x]) second_sum = sum([0.5 * i * xi for i, xi in enumerate(x, start=1)]) ** 2 third_sum = sum([0.5 * i * xi for i, xi in enumerate(x, start=1)]) ** 4 return first_sum + second_sum + third_sum def levy(x): """Levy function""" w = [] for i in range(len(x)): wi = 1 + (x[i] - 1) / 4 w.append(wi) term1 = (np.sin(np.pi * w[0])) ** 2 term3 = ((w[-1] - 1) ** 2) * (1 + (np.sin(2 * np.pi * w[-1]) ** 2)) summation = 0 for wd in range(len(w) - 1): wi = w[wd] new_term = ((wi - 1) ** 2) * (1 + 10 * (np.sin(np.pi * wi + 1)) ** 2) summation += new_term return term1 + summation + term3 def michalewicz(x): """Michalewicz function""" m = 10 summation = 0 for i, xi in enumerate(x): new_i = i + 1 summation += np.sin(xi) * ((np.sin((new_i * xi ** 2) / np.pi)) ** (2 * m)) return -summation def perm_zero_d_beta(x): """Perm Function with beta set to zero""" pass # Implementation omitted due to complexity and space constraints. def rotated_hyper_ellipsoid(x): """Rotated Hyper-Ellipsoid function""" result = 0 for i in range(len(x)): for j in range(i + 1): result += x[j] ** 2 return result def trid(x): """Trid function""" n = len(x) s1 = sum([(x[i] - 1) ** 2 for i in range(n)]) s2 = sum([x[i] * x[i - 1] for i in range(1, n)]) return s1 - s2 def bohachevsky_one(x): """Bohachevsky N.1 function""" return x[0] ** 2 + 2 * x[1] ** 2 - 0.3 * np.cos(3 * np.pi * x[0]) - \ 0.4 * np.cos(4 * np.pi * x[1]) + 0.7 def easom(x): """Easom function""" return (-np.cos(x[0]) * np.cos(x[1]) * np.exp(-(x[0] - np.pi) ** 2 - (x[1] - np.pi) ** 2)) def beale(x): """Beale function""" return ((1.5 - x[0] + x[0] * x[1]) ** 2 + (2.25 - x[0] + x[0] * x[1] ** 2) ** 2 + (2.625 - x[0] + x[0] * x[1] ** 3) ** 2) def matyas(x): """Matyas function""" return 0.26 * (x[0] ** 2 + x[1] ** 2) - 0.48 * x[0] * x[1] def levi_n13(x): """Levi No.13 function""" sin3_x1 = (np.sin(3 * np.pi * x[0])) ** 2 sin2_x2 = (np.sin(2 * np.pi * x[1])) ** 2 w = 1 + (x[0] - 1) ** 2 * (1 + sin3_x1) z = (x[1] - 1) ** 2 * (1 + sin2_x2) y = (x[0] - 1) ** 2 return w + z + y def three_hump_camel(x): """Three-hump camel function""" return 2 * x[0] ** 2 - 1.05 * x[0] ** 4 + x[0] ** 6 / 6 + x[0] * x[1] + x[1] ** 2 def six_hump_camel(x): """Six-hump camel back function""" return ( (4 - 2.1 * x[0] ** 2 + x[0] ** 4 / 3) * x[0] ** 2 + x[0] * x[1] + 4 * (x[1] ** 2 - 1) * x[1] ** 2 ) def powell_singular_function(x): """Powell singular function""" f1 = (x[0] + 10 * x[1]) ** 2 f2 = 5 * (x[2] - x[3]) ** 2 f3 = (x[1] - 2 * x[2]) ** 4 f4 = 10 * (x[0] - x[3]) ** 4 return f1 + f2 + f3 + f4 def damavandi(x): """Damavandi function""" t1 = abs( np.sin(np.pi * (x[0] - 2)) * np.sin(np.pi * (x[1] - 2)) / (np.pi ** 2 * (x[0] - 2) * (x[1] - 2)) ) ** 5 t2 = 1 - 0.5 * np.exp(-sum((x - 2) ** 2)) return t1 * t2 def shubert(x): """Shubert function""" prod = 1 for i in range(len(x)): inner_sum = 0 for j in range(1, 6): inner_sum += j * np.cos((j + 1) * x[i] + j) prod *= inner_sum return prod def hansen(x): """Hansen function""" outer = 0 for i in range(3): inner = 0 for j in range(5): inner += (j + 1) * np.