## 4 Applications in Optimization
### Particle Swarm Optimization (PSO)
#### 4.1.3 Assessment of Enhanced PSO Algorithm with Lozi Chaotic Map
With increasing population size, DE was outperformed by PSO even though both algorithms performed the same amount of CF evaluations. Widening the scope of their investigations, in [70] Zelinka's team explored two other chaotic maps: first, the dissipative standard map [71]  
\[\left\{ \begin{array}{l}x_{n+1}=x_{n+1}+y_{n+1}\ \ (mod\ 2\pi),\\ y_{n+1}=by_{n}+ksinx_{n}\ \ (mod\ 2\pi),\end{array} \right. \tag{45}\]  
with \(b=0.6\) and \(k=8.8\), and second, the Arnold's Cat map [72]  
\[\left\{ \begin{array}{l}x_{n+1}=x_{n}+y_{n}\ \ (mod\ 1),\\ y_{n+1}=x_{n}+ky_{n}\ \ (mod\ 1),\end{array} \right. \tag{46}\]  
with \(k=0.1\). They compared these to the Lozi map with \(a=1.7\) and \(b=0.5\) using the same test function (41)-(44). After a very extensive sequence of tests, they concluded that, first, the number generator based on the Arnold's Cat chaotic map does not seem to have any significant effect on the behavior or performance of the PSO algorithm with linear decreasing inertia weight. Second, the Lozi map used as the number generator seemed to have a significant positive effect on the speed of convergence of the algorithm. This may support the idea of using this modification in real-time optimization tasks on low-dimensional problems, especially because the risk of premature convergence and stagnation in higher dimensions seems to be a significant drawback of the PSO Lozi version. Finally, the PSO with the dissipative standard map seemed able to avoid premature convergence much better than any other \begin{table}
\begin{tabular}{l c c c} \hline
**Result for dim = 40** & **PSO Weight** & **PSO Lozi** & **DE** \\ \hline The worst result & **1.03 \(\times 10^{0}\)** & \(1.33\times 10^{0}\) & \(1.34\times 10^{2}\) \\ The best result & \(7.93\times 10^{-2}\) & **2.25 \(\times 10^{2}\)** & \(5.02\times 10^{1}\) \\ Median & \(3.74\times 10^{-1}\) & **2.19 \(\times 10^{1}\)** & \(7.77\times 10^{1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Results for the second De Jong function for dim = 40. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline
**Pop. number** & **30** & **50** & **75** & **100** & **150** & **200** & **300** & **400** \\ \hline PSO Weight & \(-3697.6\) & \(-3873.6\) & \(-4141.0\) & \(-4255.8\) & \(-4329.6\) & \(-4866.4\) & \(-5316.4\) & \(-5377.7\) \\ PSO Lozi & \(-4340.1\) & \(-4560.4\) & \(-5032.8\) & \(-5241.8\) & \(-**5802.0\)** & \(-**5998.1\) & \(-**6174.6**\) & \(-**6225.6**\) \\ DE & \(-\)**6100.9** & \(-\)**5737.7** & \(-\)**5649.1** & \(-\)**5500.0** & \(-5635.6\) & \(-5651.5\) & \(-5673.3\) & \(-5651.23\) \\ \hline \end{tabular}
\end{table} 
Table 2: Mean value for thirty runs with Schwefel’s function, dim = 20, and generations = 200.  
version in this study. However, a drawback was its very slow speed of convergence; in other terms, a higher number of generations were needed to find the optimal value, or at least a satisfactory solution. Finally, based on these conclusions, in [73] they found a new way to improve PSO based on the dissipative standard map and Lozi map by using them sequentially.