Enhanced Particle Swarm Optimization (PSO) with Python

Implemented fully documented Particle Swarm Optimization (PSO) algorithm in Python which includes a basic model along with few advanced features such as updating inertia weight, cognitive, social learning coefficients and maximum velocity of the particle.

Dependencies

• Numpy
• matplotlib

Utilities

Once the installation is finished (download or cloning), go the pso folder and follow the below simple guidelines to execute PSO effectively (either write the code in command line or in a python editor).

>>> from pso import PSO

Next, a fitness function (or cost function) is required. I have included four different fitness functions for example purposes namely fitness_1, fitness_2, fitness_3, and fitness_4.

Fitness-1 (Himmelblau's Function)

Minimize: f(x) = (x² + y - 11)² + (x + y² - 7)²

Optimum solution: x = 3 ; y = 2

Fitness-2 (Booth's Function)

Minimize: f(x) = (x + 2y - 7)² + (2x + y - 5)²

Optimum solution: x = 1 ; y = 3

Fitness-3 (Beale's Function)

Minimize: f(x) = (1.5 - x - xy)² + (2.25 - x + xy²)² + (2.625 - x + xy³)²

Optimum solution: x = 3 ; y = 0.5

Fitness-4

Maximize: f(x) = 2xy + 2x - x² - 2y²

Optimum solution: x = 2 ; y = 1

>>> from fitness import fitness_1, fitness_2, fitness_3, fitness_4

Now, if you want, you can provide an initial position X₀ and bound value for all the particles (not mandatory) and optimize (minimize or maximize) the fitness function using PSO:

NOTE: a bool variable min=True (default value) for MINIMIZATION PROBLEM and min=False for MAXIMIZATION PROBLEM

>>> PSO(fitness=fitness_1, X0=[1,1], bound=[(-4,4),(-4,4)]).execute()

You will see the following similar output:

OPTIMUM SOLUTION
  > [3.0000078, 1.9999873]

OPTIMUM FITNESS
  > 0.0

When fitness_4 is used, observe that min=False since it is a Maximization problem.

>>> PSO(fitness=fitness_4, X0=[1,1], bound=[(-4,4),(-4,4)], min=False).execute()

You will see the following similar output:

OPTIMUM SOLUTION
  > [2.0, 1.0]

OPTIMUM FITNESS
  > 2.0

Incase you want to print the fitness value for each iteration, then set verbose=True (here Tmax=50 is the maximum iteration)

>>> PSO(fitness=fitness_2, Tmax=50, verbose=True).execute()

You will see the following similar output:

Iteration:   0  | best global fitness (cost): 18.298822
Iteration:   1  | best global fitness (cost): 1.2203953
Iteration:   2  | best global fitness (cost): 0.8178153
Iteration:   3  | best global fitness (cost): 0.5902262
Iteration:   4  | best global fitness (cost): 0.166928
Iteration:   5  | best global fitness (cost): 0.0926638
Iteration:   6  | best global fitness (cost): 0.0926638
Iteration:   7  | best global fitness (cost): 0.0114517
Iteration:   8  | best global fitness (cost): 0.0114517
Iteration:   9  | best global fitness (cost): 0.0114517
Iteration:   10 | best global fitness (cost): 0.0078867
Iteration:   11 | best global fitness (cost): 0.0078867
Iteration:   12 | best global fitness (cost): 0.0078867
Iteration:   13 | best global fitness (cost): 0.0078867
Iteration:   14 | best global fitness (cost): 0.0069544
Iteration:   15 | best global fitness (cost): 0.0063058
Iteration:   16 | best global fitness (cost): 0.0063058
Iteration:   17 | best global fitness (cost): 0.0011039
Iteration:   18 | best global fitness (cost): 0.0011039
Iteration:   19 | best global fitness (cost): 0.0011039
Iteration:   20 | best global fitness (cost): 0.0011039
Iteration:   21 | best global fitness (cost): 0.0007225
Iteration:   22 | best global fitness (cost): 0.0005875
Iteration:   23 | best global fitness (cost): 0.0001595
Iteration:   24 | best global fitness (cost): 0.0001595
Iteration:   25 | best global fitness (cost): 0.0001595
Iteration:   26 | best global fitness (cost): 0.0001595
Iteration:   27 | best global fitness (cost): 0.0001178
Iteration:   28 | best global fitness (cost): 0.0001178
Iteration:   29 | best global fitness (cost): 0.0001178
Iteration:   30 | best global fitness (cost): 0.0001178
Iteration:   31 | best global fitness (cost): 0.0001178
Iteration:   32 | best global fitness (cost): 0.0001178
Iteration:   33 | best global fitness (cost): 0.0001178
Iteration:   34 | best global fitness (cost): 0.0001178
Iteration:   35 | best global fitness (cost): 0.0001178
Iteration:   36 | best global fitness (cost): 0.0001178
Iteration:   37 | best global fitness (cost): 2.91e-05
Iteration:   38 | best global fitness (cost): 1.12e-05
Iteration:   39 | best global fitness (cost): 1.12e-05
Iteration:   40 | best global fitness (cost): 1.12e-05
Iteration:   41 | best global fitness (cost): 1.12e-05
Iteration:   42 | best global fitness (cost): 1.12e-05
Iteration:   43 | best global fitness (cost): 1.12e-05
Iteration:   44 | best global fitness (cost): 1.12e-05
Iteration:   45 | best global fitness (cost): 1.12e-05
Iteration:   46 | best global fitness (cost): 1.12e-05
Iteration:   47 | best global fitness (cost): 2.4e-06
Iteration:   48 | best global fitness (cost): 2.4e-06
Iteration:   49 | best global fitness (cost): 2.4e-06
Iteration:   50 | best global fitness (cost): 2.4e-06

OPTIMUM SOLUTION
  > [1.0004123, 2.9990281]

OPTIMUM FITNESS
  > 2.4e-06

Now, incase you want to plot the fitness value for each iteration, then set plot=True (here Tmax=50 is the maximum iteration)

>>> PSO(fitness=fitness_2, Tmax=50, plot=True).execute()

You will see the following similar output:

OPTIMUM SOLUTION
  > [1.0028365, 2.9977422]

OPTIMUM FITNESS
  > 1.45e-05

Finally, in case you want to use the advanced features as mentioned above (say you want to update the weight inertia parameter w), simply use update_w=True and thats it. Similarly you can use update_c1=True (to update individual cognitive parameter c1), update_c2=True (to update social learning parameter c2), and update_vmax=True (to update maximum limited velocity of the particle vmax)

>>> PSO(fitness=fitness_1, update_w=True, update_c1=True).execute()

References:

[1] Almeida, Bruno & Coppo leite, Victor. (2019). Particle swarm optimization: a powerful technique for solving engineering problems. 10.5772/intechopen.89633.

[2] He, Yan & Ma, Wei & Zhang, Ji. (2016). The parameters selection of pso algorithm influencing on performance of fault diagnosis. matec web of conferences. 63. 02019. 10.1051/matecconf/20166302019.

[3] Clerc, M., and J. Kennedy. The particle swarm — explosion, stability, and convergence in a multidimensional complex space. ieee transactions on evolutionary computation 6, no. 1 (february 2002): 58–73.

[4] Y. H. Shi and R. C. Eberhart, “A modified particle swarm optimizer,” in proceedings of the ieee international conferences on evolutionary computation, pp. 69–73, anchorage, alaska, usa, may 1998.

[5] G. Sermpinis, K. Theofilatos, A. Karathanasopoulos, E. F. Georgopoulos, & C. Dunis, Forecasting foreign exchange rates with adaptive neural networks using radial-basis functions and particle swarm optimization, european journal of operational research.

[6] Particle swarm optimization (pso) visually explained (https://towardsdatascience.com/particle-swarm-optimization-visually-explained-46289eeb2e14)

[7] Rajib Kumar Bhattacharjya, Introduction to Particle Swarm Optimization (http://www.iitg.ac.in/rkbc/ce602/ce602/particle%20swarm%20algorithms.pdf)