## 6 Experimental Results
### Experimental Setup  
In our experiments, our implementations of PSO will be used to minimize the following functions, taken from [39]:  
* Sphere Model: \(f_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\)
* [PERSON]'s Problem \(2.22\): \(f_{2}(x)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|\)
* [PERSON]'s Problem \(1.2\): \(f_{3}(x)=\sum_{i=1}^{n}(\sum_{j=1}^{i}x_{j})^{2}\)
* [PERSON]'s Problem \(2.21\): \(f_{4}(x)=max_{i}\{|x_{i}|,1\leq i\leq n\}\)
* Generalized Rosenbrock's Function: \(f_{5}(x)=\sum_{i=1}^{n-1}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\)
* Step Function: \(f_{6}(x)=\sum_{i=1}^{n}(|x_{i}+0.5|)^{2}\)
* Generalized Schwefel's Problem \(2.26\): \(f_{8}(x)=\sum_{i=1}^{n}-x_{i}\sin(\sqrt{|x_{i}|})\)
* Generalized Rastrigin's Function: \(f_{9}(x)=\sum_{i=1}^{n}[x_{i}^{2}-10\cos(2\pi x_{i})+10]\)
* Ackley's Function: \(f_{10}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}\right)- \exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos 2\pi x_{i}\right)+20+e\)
* Generalized Griewank Function: \(f_{11}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos\left(\frac {x_{i}}{\sqrt{i}}\right)+1\)
* Generalized Penalized Function I: \(f_{12}(x)=\frac{\pi}{n}\Big{\{}10\sin^{2}(\pi y_{i})+\sum_{i=1}^{n-1}(y_{i}- 1)^{2}[1+10\sin^{2}(\pi y_{i+1})]+(y_{n}-1)^{2}\Big{\}}+\sum_{i=1}^{n}u(x_{i},10,100,4)\), \(y_{i}=1+\frac{1}{4}(x_{i}+1)\), \(u(x_{i},a,k,m)=\left\{\begin{array}{ll}k(x_{i}-a)^{m},&x_{i}>a\\ 0,&-a\leq x_{i}\leq a\\ k(-x_{i}-a)^{m},&x_{i}<-a\end{array}\right.\)
* Generalized Penalized Function II: \(f_{13}(x)=0.1\Big{\{}\sin^{2}(3\pi x_{1})+\sum_{i=1}^{n-1}(x_{i}-1)^{2}[1+ \sin^{2}(3\pi x_{i+1})]+(x_{n}-1)^{2}[1+sin^{2}(2\pi x_{n})]\Big{\}}+\sum_{i=1 }^{n}u(x_{i},5,100,4)\)
* Shekel's Foxholes Function: \(f_{14}(x)=\left[\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+\sum_{i=1}^{2}(x_{i}- a_{ij})^{6}}\right]\)
* Kowalik's Function: \(f_{15}(x)=\sum_{i=1}^{11}\left[a_{i}-\frac{x_{1}(b_{i}^{2}+b_{i}x_{2})}{b_{i}^{2 }+b_{i}x_{3}+x_{4}}\right]^{2}\)
* Six-Hump Camel-Back Function: \(f_{16}(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+ 4x_{2}^{4}\)
* Branin Function: \(f_{17}(x)=\left(x_{2}-\frac{5.1}{4\pi^{2}}x_{1}^{2}+\frac{5}{\pi}x_{1}-6 \right)^{2}+10\Big{(}1-\frac{1}{8\pi}\Big{)}\cos x_{1}+10\)  
The most relevant features of these functions are depicted in Table 1. For each function, we show the number of dimensions, the boundaries of the solutions space, the value of the optimal solution, and the asymmetric initialization range used in the experiments. The first six functions are relatively simple, whereas finding the minimum value of functions \(f_{8}-f_{13}\) is harder. The reason is that there are more local minima in these functions. In fact, the number of local minima is exponential with respect to the number of dimensions of these problems. Functions \(f_{14}-f_{17}\) are low-dimensional functions which have only a few local minima.  
In our experiments, values \(\omega\), \(\phi p\), and \(\phi r\) are set as proposed in [29]. For each function, the number of iterations is the same as the number of algorithm iterations performed in [39], while the number of particles is \(100\). Following the asymmetric initialization approach given in [40], particles are asymmetrically distributed for all problems but \(f_{8}\), which is appointed as specially hard for using an asymmetric initialization in that work. In this special case we use an homogeneous distribution, as it is done in [40].