Particle-Swarm-Optimisation

A simple Particle Swarm Optimisation algorithm, finding the minimum of several functions in the N-dimensional space.

Benchmark Functions

Several benchmark functions were used to evaluate the effectiveness of the algorithm.

Ackley

$$ f_{Ackley}(x) = - 20 \times e^{- 0.2 \times \sqrt{\frac{1}{n} \sum_{i=1}^{n} {x_{i}^{2}}}} \times e^{\frac{1}{n} \sum_{i = 1}^{n} \cos{2\pi x_{i}}} + 20 + e $$

Ackley Particle Swarm Optimisation in the 2-dimensional space

Griwank

$$ f_{Griewank}(x) = 1 + \frac{1}{4000} \sum_{i = 1}^{n} {x_{i}^{2}} - \prod_{i = 1}^{n} {\cos \frac{x_{i}}{\sqrt{i}}} $$

Griewank Particle Swarm Optimisation in the 2-dimensional space

Michalewicz

$$ f_{Michalewicz}(x) = - \sum_{i = 1}{n} \sin {x_{i}} \times {(\sin {\frac {ix_{i}^{2}}{\pi}})}^{2m} $$

, $where\ m = 10$

Michalewicz Particle Swarm Optimisation in the 2-dimensional space

Rastrigin

$$ f_{Rastrigin}(x) = a \times n + \sum_{i = 1}^{n} {x_{i}^{2} - a \cos {2 \pi x_{i}}} $$

, $where\ a = 10$

Rastrigin Particle Swarm Optimisation in the 2-dimensional space

Rosenbrock

$$ f_{Rosenbrock}(x) = \sum_{i=1}^{n-1} (100(x_{i+1} - x_{i}^{2}) + (x_{i} - 1)^{2}) $$

Rosenbrock Particle Swarm Optimisation in the 2-dimensional space

Sphere

$$ f_{Sphere}(x) = \sum_{i=1}^{n} {x_{i}^{2}} $$

Sphere Particle Swarm Optimisation in the 2-dimensional space

Evaluate

The algorithm is evaluated against two criteria:

1. Average Loss

Average Loss is calculated as follows $\overline{f(p)} - f(global_{min})$ where $p$ is a particle's current position and $global_{min}$ is the technical global minimum of the function.

The graphs shown are all in the 10 Dimensional Space.

Ackley

Ackley Average Loss in the 10 Dimensional Space over Iterations

Griewank

Griewank Average Loss in the 10 Dimensional Space over Iterations

Michalewicz

Michalewicz Average Loss in the 10 Dimensional Space over Iterations

Rastrigin

Rastrigin Average Loss in the 10 Dimensional Space over Iterations

Rosenbrock

Rosenbrock Average Loss in the 10 Dimensional Space over Iterations

Sphere

Sphere Average Loss in the 10 Dimensional Space over Iterations

2. Number of Successes

The number of successes is evaluated base on how close the global best position of the terminated particle swarm optimisation is to the technical global minimum of the function. The condition is rather harsh, having used the NumPy allclose function to evaluate the closeness of the final position to the actual position. The result is shown in the graph below: