Fitness functions

1) Rastrigin function

Rastrigin function is a non-convex function and is often used as a performance test problem for optimization algorithms.

function equation:

Fig1: Rastrigin function for 2 variables

For an optimization algorithm, rastrigin function is a very challenging one. Its complex behavior causes optimization algorithms to often be stuck at local minima. Having a lot of cosine oscillations on the plane introduces the complex behavior to this function.

2) Sphere function

Sphere function is a standard function for evaluating the performance of an optimization algorithm.

function equation:

Fig 2: Sphere function for 2 variables

Implementation of Whale Optimization Algorithm

Previous article Whale optimization algorithm (WOA) talked about the inspiration of whale optimization, its mathematical modeling and algorithm. In this article we will implement a whale optimization algorithm (WOA) for two fitness functions 1) Rastrigin function 2) Sphere function The algorithm will run for a predefined number of maximum iterations and will try to find the minimum value of these fitness functions.

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Python3 # python implementation of whale optimization algorithm (WOA)# 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; # ------------------------- # whale classclass 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 = 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_whales = 50max_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 functionprint("\nBegin whale optimization algorithm 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_whales = 50max_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")...

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#Artificial Intelligence #python #AI-ML-DS #Machine Learning #Machine Learning #python

Choice of hyper-parameters

Fitness functions

1) Rastrigin function

function equation:

2) Sphere function

function equation:

Implementation of Whale Optimization Algorithm

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