# Implementation of Grey Wolf Optimization (GWO) Algorithm

• Last Updated : 27 Feb, 2022

Previous article Grey wolf optimization- Introduction talked about inspiration of grey wolf optimization, and its mathematical modelling and algorithm. In this article we will implement grey wolf optimization (PSO) for two fitness functions – Rastrigin function and Sphere function. The aim of Grey wolf optimization algorithm is to find minimize of fitness function.

### Fitness Functions:

1) Rastrigin function: Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms.

Function equation:  Rastrigin Function is one of the most challenging functions for an optimization problem. Having a lot of cosine oscillations on the plane introduces a myriad of local minimums in which particles can get stuck.

2) Sphere function: Sphere function is used as a performance test problem for optimization algorithms.

Function equation:   Figure2: Sphere function of two variables

### Choice of hyper-parameters

• Parameters of problem:
• Number of dimensions (d) = 3
• Lower bound (minx) = -10.0
• Upper bound (maxx) = 10.0
• Hyperparameters of the algorithm:
• Number of grey wolves (N) = 50
• Maximum number of iterations (max_iter) = 100
• Inputs:
• Fitness function
• Problem parameters ( mentioned above)
• Population size (N) and Maximum number of iterations  (max_iter)
• Algorithm Specific hyperparameters ( None in grey wolf algorithm)

### Pseudocode:

Step1: Randomly initialize Grey wolf population of N particles Xi ( i=1, 2, …, n)
Step2: Calculate the fitness value of each individuals
sort grey wolf population based on fitness values
alpha_wolf = wolf with least fitness value
beta_wolf  = wolf with second least fitness value
gamma_wolf = wolf with third least fitness value
Step 3: For Iter in range(max_iter):  # loop max_iter times
calculate the value of a
a = 2*(1 - Iter/max_iter)
For i in range(N):  # for each wolf
a. Compute the value of A1, A2, A3 and C1, C2, C3
A1 = a*(2*r1 -1), A2 = a*(2*r2 -1), A3 = a*(2*r3 -1)
C1 = 2*r1, C2 = 2*r2, C3 = 2*r3

b. Computer X1, X2, X3
X1 = alpha_wolf.position -
A1*abs(C1*alpha_wolf_position - ith_wolf.position)
X2 = beta_wolf.position -
A2*abs(C2*beta_wolf_position - ith_wolf.position)
X3 = gamma_wolf.position -
A3*abs(C3*gamma_wolf_position - ith_wolf.position)

c. Compute new solution and it's fitness
Xnew = (X1 + X2 + X3) / 3
fnew = fitness( Xnew)

d. Update the ith_wolf greedily
if( fnew < ith_wolf.fitness)
ith_wolf.position = Xnew
ith_wolf.fitness = fnew
End-for

# compute new alpha, beta and gamma
sort grey wolf population based on fitness values
alpha_wolf = wolf with least fitness value
beta_wolf  = wolf with second least fitness value
gamma_wolf = wolf with third least fitness value
End -for
Step 4: Return best wolf in the population

## Python3

 # python implementation of Grey wolf optimization (GWO)# minimizing rastrigin and sphere function  import randomimport math    # cos() for Rastriginimport copy    # array-copying convenienceimport sys     # max float  #-------fitness functions--------- # rastrigin functiondef 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 functiondef fitness_sphere(position):    fitness_value = 0.0    for i in range(len(position)):        xi = position[i]        fitness_value += (xi*xi);    return fitness_value;#-------------------------  # wolf classclass wolf:  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   # grey wolf optimization (GWO)def gwo(fitness, max_iter, n, dim, minx, maxx):    rnd = random.Random(0)     # create n random wolves    population = [ wolf(fitness, dim, minx, maxx, i) for i in range(n)]     # On the basis of fitness values of wolves    # sort the population in asc order    population = sorted(population, key = lambda temp: temp.fitness)     # best 3 solutions will be called as    # alpha, beta and gaama    alpha_wolf, beta_wolf, gamma_wolf = copy.copy(population[: 3])      # main loop of gwo    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" % alpha_wolf.fitness)         # linearly decreased from 2 to 0        a = 2*(1 - Iter/max_iter)         # updating each population member with the help of best three members        for i in range(n):            A1, A2, A3 = a * (2 * rnd.random() - 1), a * (              2 * rnd.random() - 1), a * (2 * rnd.random() - 1)            C1, C2, C3 = 2 * rnd.random(), 2*rnd.random(), 2*rnd.random()             X1 = [0.0 for i in range(dim)]            X2 = [0.0 for i in range(dim)]            X3 = [0.0 for i in range(dim)]            Xnew = [0.0 for i in range(dim)]            for j in range(dim):                X1[j] = alpha_wolf.position[j] - A1 * abs(                  C1 * alpha_wolf.position[j] - population[i].position[j])                X2[j] = beta_wolf.position[j] - A2 * abs(                  C2 *  beta_wolf.position[j] - population[i].position[j])                X3[j] = gamma_wolf.position[j] - A3 * abs(                  C3 * gamma_wolf.position[j] - population[i].position[j])                Xnew[j]+= X1[j] + X2[j] + X3[j]                         for j in range(dim):                Xnew[j]/=3.0                         # fitness calculation of new solution            fnew = fitness(Xnew)             # greedy selection            if fnew < population[i].fitness:                population[i].position = Xnew                population[i].fitness = fnew                         # On the basis of fitness values of wolves        # sort the population in asc order        population = sorted(population, key = lambda temp: temp.fitness)         # best 3 solutions will be called as        # alpha, beta and gaama        alpha_wolf, beta_wolf, gamma_wolf = copy.copy(population[: 3])                 Iter+= 1    # end-while     # returning the best solution    return alpha_wolf.position           #----------------------------  # Driver code for rastrigin function print("\nBegin grey wolf optimization on rastrigin function\n")dim = 3fitness = 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_particles = 50max_iter = 100 print("Setting num_particles = " + str(num_particles))print("Setting max_iter    = " + str(max_iter))print("\nStarting GWO algorithm\n")   best_position = gwo(fitness, max_iter, num_particles, dim, -10.0, 10.0) print("\nGWO 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 GWO for rastrigin\n")  print()print()  # Driver code for Sphere functionprint("\nBegin grey wolf optimization on sphere function\n")dim = 3fitness = 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_particles = 50max_iter = 100 print("Setting num_particles = " + str(num_particles))print("Setting max_iter    = " + str(max_iter))print("\nStarting GWO algorithm\n")   best_position = gwo(fitness, max_iter, num_particles, dim, -10.0, 10.0) print("\nGWO 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 GWO for sphere\n")

### Output:

Begin grey wolf optimization 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_particles = 50
Setting max_iter    = 100

Starting GWO algorithm

Iter = 10 best fitness = 2.996
Iter = 20 best fitness = 2.749
Iter = 30 best fitness = 0.470
Iter = 40 best fitness = 0.185
Iter = 50 best fitness = 0.005
Iter = 60 best fitness = 0.001
Iter = 70 best fitness = 0.001
Iter = 80 best fitness = 0.001
Iter = 90 best fitness = 0.000

GWO completed

Best solution found:
['0.000706', '-0.000746', '-0.000526']
fitness of best solution = 0.000264

End GWO for rastrigin

Begin grey wolf optimization on sphere function

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

Starting GWO algorithm

Iter = 10 best fitness = 0.001
Iter = 20 best fitness = 0.001
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

GWO completed

Best solution found:
['-0.000064', '0.000879', '-0.000934']
fitness of best solution = 0.000002

End GWO for sphere

### References:

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