Editing Particle swarm optimization (section)

==== Accelerated Particle Swarm Optimization ====
Another simpler variant is the accelerated particle swarm optimization (APSO),<ref>X. S. Yang, S. Deb and S. Fong, [https://arxiv.org/abs/1203.6577 Accelerated particle swarm optimization and support vector machine for business optimization and applications], NDT 2011, Springer CCIS 136, pp. 53-66 (2011).</ref> which also does not need to use velocity and can speed up the convergence in many applications. A simple demo code of APSO is available.<ref>{{Cite web | url=http://www.mathworks.com/matlabcentral/fileexchange/?term=APSO | title=Search Results: APSO - File Exchange - MATLAB Central}}</ref>

In this variant of PSO one dispences with both the particle's velocity and the particle's best position. The particle position is updated according to the following rule,
:<math>
\vec x_i \leftarrow (1-\beta)\vec x_i + \beta \vec g + \alpha L \vec u \,,
</math>
where <math>\vec u</math> is a random uniformly distributed vector, <math>L</math> is the typical length of the problem at hand, and <math>\beta\sim 0.1-0.7</math> and <math>\alpha\sim 0.1-0.5</math> are the parameters of the method. As a refinement of the method one can decrease <math>\alpha</math> with each iteration, <math>\alpha_n=\alpha_0\gamma^n</math>, where <math>n</math> is the number of the iteration and <math>0 < \gamma < 1</math> is the decrease control parameter.