Particle swarm optimization: the simplest what and how

While there exist many introductory materials on Particle Swarm Optimization (PSO), it is best to have an intuitive example, simple, understandable at first reading, and illustrative. I personally do not like those examples that try to bring all intricacies together. Here comes the simplest example I guess: to find a minimum for the function:

It is know that the optimal solution is found when x=-1, and the minimum is f(x)=0;

The below video demonstrate the results of finding this optimal solution using PSO based approach:

As you can see, after some initial random movements of the design variables, the design variable gradually converges to x=-1 (the black curve) and the optimal value reaches 0.

The basic procedures to implement this PSO based approach are as follows:

[1] Create a collection of particles, each particle represents an instance of the design variable. In this case, the design variable is a scalar-valued single parameter:

[2] For these particles, evaluate their performances:

[3] Find the best of these particles, which generates the minimum errors

[4] The minimum value got so far is both the local best and the global best, keep them down for later use:

[5] For each particles, move them according to the formula:

For details of these parameters and their physical meanings, Google “Particle Swarm Optimization”, a quick view of the result goes here. 

[6] Terminate when the some conditions are met:

Here, RunMoveOnce() refers to the process described in [5].

The Matlab files can be download it below. Enjoy!

v0.1  | v0.2  (Nov, 2010)