鲸鱼优化算法的实现

上一篇鲸鱼优化算法（WOA）谈到了鲸鱼优化的灵感，它的数学建模和算法。在本文中，我们将为两个适应度函数实现鲸鱼优化算法 (WOA) 1) Rastrigin函数2) 球体函数该算法将运行预定义的最大迭代次数，并尝试找到这些适应度函数的最小值。

健身功能

1) 拉斯特里金函数

Rastrigin函数是非凸函数，常被用作优化算法的性能测试问题。

函数方程：

$f(x_1 \cdots x_n) = 10n + \sum_{i=1}^n (x_i^2 -10cos(2\pi x_i))$

$\text{minimum at }f(0, \cdots, 0) = 0$

图 1：2 个变量的 Rastrigin函数

对于优化算法，rastrigin函数是一个非常具有挑战性的函数。它的复杂行为导致优化算法经常停留在局部最小值。在平面上有很多余弦振荡会为该函数引入复杂的行为。

2) 球面函数

球面函数是用于评估优化算法性能的标准函数。

函数方程：

$f(x_1 \cdots x_n) = \sum_{i=1}^n x_i^2$

$\text{minimum at }f(0, \cdots, 0) = 0$

图 2：2 个变量的球体函数

超参数的选择

问题参数：

• 维数 ( d ) = 3
• 下限 ( minx ) = -10.0
• 上限 ( maxx ) = 10.0

算法的超参数：

• 粒子数 ( N ) = 50
• 最大迭代次数 ( max_iter ) = 100
• 螺旋系数 ( b ) = 1

输入

• 健身函数
• 问题参数（上面提到的）
• 总体大小 ( N ) 和最大迭代次数 ( max_iter )
• 算法特定的超参数b

算法

鲸鱼优化的算法和数学方程已经在上一篇文章中描述过了。

执行

Python3

# python implementation of whale optimization algorithm (WOA)
# minimizing rastrigin and sphere function
 
 
import random
import math  # cos() for Rastrigin
import copy  # array-copying convenience
import sys  # max float
 
 
# -------fitness functions---------
 
# rastrigin function
def fitness_rastrigin(position):
    fitness_value = 0.0
    for i in range(len(position)):
        xi = position[i]
        fitness_value += (xi * xi) - (10 * math.cos(2 * math.pi * xi)) + 10
    return fitness_value
 
 
# sphere function
def fitness_sphere(position):
    fitness_value = 0.0
    for i in range(len(position)):
        xi = position[i]
        fitness_value += (xi * xi);
    return fitness_value;
 
 
# -------------------------
 
 
# whale class
class whale:
    def __init__(self, fitness, dim, minx, maxx, seed):
        self.rnd = random.Random(seed)
        self.position = [0.0 for i in range(dim)]
 
        for i in range(dim):
            self.position[i] = ((maxx - minx) * self.rnd.random() + minx)
 
        self.fitness = fitness(self.position)  # curr fitness
 
 
# whale optimization algorithm(WOA)
def woa(fitness, max_iter, n, dim, minx, maxx):
    rnd = random.Random(0)
 
    # create n random whales
    whalePopulation = [whale(fitness, dim, minx, maxx, i) for i in range(n)]
 
    # compute the value of best_position and best_fitness in the whale Population
    Xbest = [0.0 for i in range(dim)]
    Fbest = sys.float_info.max
 
    for i in range(n):  # check each whale
        if whalePopulation[i].fitness < Fbest:
            Fbest = whalePopulation[i].fitness
            Xbest = copy.copy(whalePopulation[i].position)
 
    # main loop of woa
    Iter = 0
    while Iter < max_iter:
 
        # after every 10 iterations
        # print iteration number and best fitness value so far
        if Iter % 10 == 0 and Iter > 1:
            print("Iter = " + str(Iter) + " best fitness = %.3f" % Fbest)
 
        # linearly decreased from 2 to 0
        a = 2 * (1 - Iter / max_iter)
        a2=-1+Iter*((-1)/max_iter)
 
        for i in range(n):
            A = 2 * a * rnd.random() - a
            C = 2 * rnd.random()
            b = 1
            l = (a2-1)*rnd.random()+1;
            p = rnd.random()
 
            D = [0.0 for i in range(dim)]
            D1 = [0.0 for i in range(dim)]
            Xnew = [0.0 for i in range(dim)]
            Xrand = [0.0 for i in range(dim)]
            if p < 0.5:
                if abs(A) > 1:
                    for j in range(dim):
                        D[j] = abs(C * Xbest[j] - whalePopulation[i].position[j])
                        Xnew[j] = Xbest[j] - A * D[j]
                else:
                    p = random.randint(0, n - 1)
                    while (p == i):
                        p = random.randint(0, n - 1)
 
                    Xrand = whalePopulation[p].position
 
                    for j in range(dim):
                        D[j] = abs(C * Xrand[j] - whalePopulation[i].position[j])
                        Xnew[j] = Xrand[j] - A * D[j]
            else:
                for j in range(dim):
                    D1[j] = abs(Xbest[j] - whalePopulation[i].position[j])
                    Xnew[j] = D1[j] * math.exp(b * l) * math.cos(2 * math.pi * l) + Xbest[j]
 
            for j in range(dim):
                whalePopulation[i].position[j] = Xnew[j]
 
        for i in range(n):
            # if Xnew < minx OR Xnew > maxx
            # then clip it
            for j in range(dim):
                whalePopulation[i].position[j] = max(whalePopulation[i].position[j], minx)
                whalePopulation[i].position[j] = min(whalePopulation[i].position[j], maxx)
 
            whalePopulation[i].fitness = fitness(whalePopulation[i].position)
 
            if (whalePopulation[i].fitness < Fbest):
                Xbest = copy.copy(whalePopulation[i].position)
                Fbest = whalePopulation[i].fitness
 
 
        Iter += 1
    # end-while
 
    # returning the best solution
    return Xbest
 
 
# ----------------------------
 
 
# Driver code for rastrigin function
 
print("\nBegin whale optimization algorithm on rastrigin function\n")
dim = 3
fitness = fitness_rastrigin
 
print("Goal is to minimize Rastrigin's function in " + str(dim) + " variables")
print("Function has known min = 0.0 at (", end="")
for i in range(dim - 1):
    print("0, ", end="")
print("0)")
 
num_whales = 50
max_iter = 100
 
print("Setting num_whales = " + str(num_whales))
print("Setting max_iter    = " + str(max_iter))
print("\nStarting WOA algorithm\n")
 
best_position = woa(fitness, max_iter, num_whales, dim, -10.0, 10.0)
 
print("\nWOA completed\n")
print("\nBest solution found:")
print(["%.6f" % best_position[k] for k in range(dim)])
err = fitness(best_position)
print("fitness of best solution = %.6f" % err)
 
print("\nEnd WOA for rastrigin\n")
 
print()
print()
 
# Driver code for Sphere function
print("\nBegin whale optimization algorithm on sphere function\n")
dim = 3
fitness = fitness_sphere
 
print("Goal is to minimize sphere function in " + str(dim) + " variables")
print("Function has known min = 0.0 at (", end="")
for i in range(dim - 1):
    print("0, ", end="")
print("0)")
 
num_whales = 50
max_iter = 100
 
print("Setting num_whales = " + str(num_whales))
print("Setting max_iter    = " + str(max_iter))
print("\nStarting WOA algorithm\n")
 
best_position = woa(fitness, max_iter, num_whales, dim, -10.0, 10.0)
 
print("\nWOA completed\n")
print("\nBest solution found:")
print(["%.6f" % best_position[k] for k in range(dim)])
err = fitness(best_position)
print("fitness of best solution = %.6f" % err)
 
print("\nEnd WOA for sphere\n")

输出

Begin whale optimization algorithm on rastrigin function

Goal is to minimize Rastrigin's function in 3 variables
Function has known min = 0.0 at (0, 0, 0)
Setting num_whales = 50
Setting max_iter    = 100

Starting WOA algorithm

Iter = 10 best fitness = 0.018
Iter = 20 best fitness = 0.000
Iter = 30 best fitness = 0.000
Iter = 40 best fitness = 0.000
Iter = 50 best fitness = 0.000
Iter = 60 best fitness = 0.000
Iter = 70 best fitness = 0.000
Iter = 80 best fitness = 0.000
Iter = 90 best fitness = 0.000

WOA completed


Best solution found:
['0.000000', '-0.000000', '-0.000000']
fitness of best solution = 0.000000

End WOA for rastrigin




Begin whale optimization algorithm on sphere function

Goal is to minimize sphere function in 3 variables
Function has known min = 0.0 at (0, 0, 0)
Setting num_whales = 50
Setting max_iter    = 100

Starting WOA algorithm

Iter = 10 best fitness = 0.130
Iter = 20 best fitness = 0.000
Iter = 30 best fitness = 0.000
Iter = 40 best fitness = 0.000
Iter = 50 best fitness = 0.000
Iter = 60 best fitness = 0.000
Iter = 70 best fitness = 0.000
Iter = 80 best fitness = 0.000
Iter = 90 best fitness = 0.000

WOA completed


Best solution found:
['0.000000', '0.000000', '-0.000000']
fitness of best solution = 0.000000

End WOA for sphere

参考：

研究论文： https : //www.sciencedirect.com/science/article/pii/S0965997816300163

作者的原始实现（在 MATLAB 中）： https : //www.mathworks.com/matlabcentral/fileexchange/55667-the-whale-optimization-algorithm