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Implementation of Particle Swarm Optimization

  • Last Updated : 31 Aug, 2021

Previous article Particle Swarm Optimization – An Overview talked about inspiration of particle swarm optimization (PSO) , it’s mathematical modelling and algorithm. In this article we will implement particle swarm optimization (PSO) 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.

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:

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

Fig1: Rastrigin function for 2 variables

For an optimization algorithm, rastrigin function is a very challenging one. Its complex behavior cause optimization algorithms to often 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: 

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

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

Fig2: Sphere function for 2 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 particles (N) = 50
  • Maximum number of iterations (max_iter) = 100
  • inertia coefficient (w) = 0.729
  • cognitive coefficient (c1) = 1.49445
  • social coefficient (c2) = 1.49445

Inputs

  • Fitness function
  • Problem parameters ( mentioned above)
  • Population size (N) and Maximum number of iterations  (max_iter)
  • Algorithm Specific hyper parameters ( w, c1, c2)

Pseudocode

The pseudocode of the particle swarm optimization is already described in the previous article. Data structures to store Swarm population, as well as a data structure to store data specific to individual particle, were also discussed.

Implementation

Python3




# python implementation of particle swarm optimization (PSO)
# 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):
  fitnessVal = 0.0
  for i in range(len(position)):
    xi = position[i]
    fitnessVal += (xi * xi) - (10 * math.cos(2 * math.pi * xi)) + 10
  return fitnessVal
 
#sphere function
def fitness_sphere(position):
    fitnessVal = 0.0
    for i in range(len(position)):
        xi = position[i]
        fitnessVal += (xi*xi);
    return fitnessVal;
#-------------------------
 
#particle class
class Particle:
  def __init__(self, fitness, dim, minx, maxx, seed):
    self.rnd = random.Random(seed)
 
    # initialize position of the particle with 0.0 value
    self.position = [0.0 for i in range(dim)]
 
     # initialize velocity of the particle with 0.0 value
    self.velocity = [0.0 for i in range(dim)]
 
    # initialize best particle position of the particle with 0.0 value
    self.best_part_pos = [0.0 for i in range(dim)]
 
    # loop dim times to calculate random position and velocity
    # range of position and velocity is [minx, max]
    for i in range(dim):
      self.position[i] = ((maxx - minx) *
        self.rnd.random() + minx)
      self.velocity[i] = ((maxx - minx) *
        self.rnd.random() + minx)
 
    # compute fitness of particle
    self.fitness = fitness(self.position) # curr fitness
 
    # initialize best position and fitness of this particle
    self.best_part_pos = copy.copy(self.position)
    self.best_part_fitnessVal = self.fitness # best fitness
 
# particle swarm optimization function
def pso(fitness, max_iter, n, dim, minx, maxx):
  # hyper parameters
  w = 0.729    # inertia
  c1 = 1.49445 # cognitive (particle)
  c2 = 1.49445 # social (swarm)
 
  rnd = random.Random(0)
 
  # create n random particles
  swarm = [Particle(fitness, dim, minx, maxx, i) for i in range(n)]
 
  # compute the value of best_position and best_fitness in swarm
  best_swarm_pos = [0.0 for i in range(dim)]
  best_swarm_fitnessVal = sys.float_info.max # swarm best
 
  # computer best particle of swarm and it's fitness
  for i in range(n): # check each particle
    if swarm[i].fitness < best_swarm_fitnessVal:
      best_swarm_fitnessVal = swarm[i].fitness
      best_swarm_pos = copy.copy(swarm[i].position)
 
  # main loop of pso
  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" % best_swarm_fitnessVal)
 
    for i in range(n): # process each particle
       
      # compute new velocity of curr particle
      for k in range(dim):
        r1 = rnd.random()    # randomizations
        r2 = rnd.random()
     
        swarm[i].velocity[k] = (
                                 (w * swarm[i].velocity[k]) +
                                 (c1 * r1 * (swarm[i].best_part_pos[k] - swarm[i].position[k])) + 
                                 (c2 * r2 * (best_swarm_pos[k] -swarm[i].position[k]))
                               
 
 
        # if velocity[k] is not in [minx, max]
        # then clip it
        if swarm[i].velocity[k] < minx:
          swarm[i].velocity[k] = minx
        elif swarm[i].velocity[k] > maxx:
          swarm[i].velocity[k] = maxx
 
 
      # compute new position using new velocity
      for k in range(dim):
        swarm[i].position[k] += swarm[i].velocity[k]
   
      # compute fitness of new position
      swarm[i].fitness = fitness(swarm[i].position)
 
      # is new position a new best for the particle?
      if swarm[i].fitness < swarm[i].best_part_fitnessVal:
        swarm[i].best_part_fitnessVal = swarm[i].fitness
        swarm[i].best_part_pos = copy.copy(swarm[i].position)
 
      # is new position a new best overall?
      if swarm[i].fitness < best_swarm_fitnessVal:
        best_swarm_fitnessVal = swarm[i].fitness
        best_swarm_pos = copy.copy(swarm[i].position)
     
    # for-each particle
    Iter += 1
  #end_while
  return best_swarm_pos
# end pso
 
 
#----------------------------
# Driver code for rastrigin function
 
print("\nBegin particle swarm optimization 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_particles = 50
max_iter = 100
 
print("Setting num_particles = " + str(num_particles))
print("Setting max_iter    = " + str(max_iter))
print("\nStarting PSO algorithm\n")
 
 
 
best_position = pso(fitness, max_iter, num_particles, dim, -10.0, 10.0)
 
print("\nPSO completed\n")
print("\nBest solution found:")
print(["%.6f"%best_position[k] for k in range(dim)])
fitnessVal = fitness(best_position)
print("fitness of best solution = %.6f" % fitnessVal)
 
print("\nEnd particle swarm for rastrigin function\n")
 
 
print()
print()
 
 
# Driver code for Sphere function
print("\nBegin particle swarm optimization 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_particles = 50
max_iter = 100
 
print("Setting num_particles = " + str(num_particles))
print("Setting max_iter    = " + str(max_iter))
print("\nStarting PSO algorithm\n")
 
 
 
best_position = pso(fitness, max_iter, num_particles, dim, -10.0, 10.0)
 
print("\nPSO completed\n")
print("\nBest solution found:")
print(["%.6f"%best_position[k] for k in range(dim)])
fitnessVal = fitness(best_position)
print("fitness of best solution = %.6f" % fitnessVal)
 
print("\nEnd particle swarm for sphere function\n")

Output:

Begin particle swarm 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 PSO algorithm

Iter = 10 best fitness = 8.463
Iter = 20 best fitness = 4.792
Iter = 30 best fitness = 2.223
Iter = 40 best fitness = 0.251
Iter = 50 best fitness = 0.251
Iter = 60 best fitness = 0.061
Iter = 70 best fitness = 0.007
Iter = 80 best fitness = 0.005
Iter = 90 best fitness = 0.000

PSO completed


Best solution found:
['0.000618', '0.000013', '0.000616']
fitness of best solution = 0.000151

End particle swarm for rastrigin function




Begin particle swarm 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 PSO algorithm

Iter = 10 best fitness = 0.189
Iter = 20 best fitness = 0.012
Iter = 30 best fitness = 0.001
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

PSO completed


Best solution found:
['0.000004', '-0.000001', '0.000007']
fitness of best solution = 0.000000

End particle swarm for sphere function

References

Research paper citation: Kennedy, J. and Eberhart, R., 1995, November. Particle swarm optimization. In Proceedings of ICNN’95-international conference on neural networks (Vol. 4, pp. 1942-1948). IEEE.

Inspiration of the implementation: https://fr.mathworks.com/matlabcentral/fileexchange/67429-a-simple-implementation-of-particle-swarm-optimization-pso-algorithm

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