## 4 Applications in Optimization
### Particle Swarm Optimization (PSO)
#### 4.1.3 Assessment of Enhanced PSO Algorithm with Lozi Chaotic Map  
In order to better evaluate the improvement conferred by the use of chaos in PSO algorithms, Pluhacek et al. [65], investigated the effect of tuning the control parameters \(a\) and \(b\) of (3) (recall that in all three above examples the parameter values were fixed to \(a=1.7\) and \(b=0.5\)). Three different benchmark functions from the IEEE's CEC 2013 benchmark suite [66] were selected for the tuning experiments. The controlling parameters of the Lozi map were set as follows:  
\[ \begin{array}{l}a:1.3-1.7,\,\text{step 0.05}\\ b:0.1-0.6,\,\text{step 0.05}\end{array} \]  
The benchmark functions were evaluated for dimension = 10, population size = 40, number of iterations = 2500, \(w_{start}=0.9\), and \(w_{end}=0.4\) according to the demands in [66]. The PSO algorithm with an adequately set Chaotic Pseudo-Random Number Generator (CPNNG) was run 100 times, and the mean results for each function and Lozi map setting were analyzed. The presented results support the claim that different settings of the chaotic map lead to significantly different CPRNG attributes, and have a great effect on the subsequent performance of the chaotic PSO algorithm. It seems that the most promising values for the parameter are \(a\in[1.4\)-\(1.5]\) and \(b\in[0.35\)-\(0.45]\). In [67,68], the authors compared the classical version of PSO with inertia weight (labelled PSO weight), PSO with Lozi map (PSO Lozi) with the sole classical value of the parameters \(a=1.7\), \(b=0.5\), and Differential Evolution (DE) with the \"DE/Rand/1/Bin\" strategy (see the next Section), for which they used four different benchmark functions in different dimensions:  
* The first De Jong function: \[f(x)=\sum_{i=1}^{dim}x_{i}^{2}\] (41)
* The second De Jong function: \[f(x)=\sum_{i=1}^{dim-1}100\Big{(}x_{i}^{2}-x_{i+1}\Big{)}^{2}+(1-x_{i})^{2}\] (42)
* The Rastrigin function: \[f(x)=10 dim\sum_{i=1}^{dim}x_{i}^{2}-10 cos(2\pi x_{i})\] (43)
* The Schwefel function: \[f(x)=\sum_{i=1}^{dim}-x_{i}sin(\sqrt{|x_{i}|})\] (44)  
The basic PSO control parameters were set based on previous experiments and the literature, as follows: population size 30, iterations/generations 10 * the dimension, and dimensions of 2, 5, 10, 20, and 40. For statistical reasons, optimization for each dimension value was repeated thirty times. The numerical results with the first De Jong function highlighted that implementation of the chaotic number generator improved the performance of the PSO algorithm. The results were better or comparable with those of differential evolution. In the case of higher dimensions (20 and 40), DE provided better results, and the differences between the results of PSO Weight and PSO Lozi were less significant. * For the second De Jong function, PSO Lozi obtained the best results in almost all tests performed with this benchmark function. The performance of both PSO algorithms was better than that of DE. In the case of Rastrigin's benchmark function, there was no significant improvement in PSO performance; however, both PSO algorithms were significantly worse than DE for dim = 40. * The results for Schwefel's function showed an increasing difference between the median values for PSO Weight and PSO Lozi, together with increasing dimension in favour of the PSO enhanced with the Lozi map. As in the previous case, both algorithms were surpassed by DE. Pluhacek et al. deepened their previous analysis in [69], enlarging the tested population size from 30 to 50, 75, 100, 150, 200, 300, and 400. For smaller population sizes, they found similar results for the first and second De Jong functions as well as for the Rastrigin and Schwefel functions (for an example, see the case dim = 40 for the Second De Jong function in Table 1). However, they found that increasing the size of the population led to significantly improved performance of both PSO algorithms with inertia weight (Table 2). From this investigation, they concluded that the population size has very significant influence on both algorithms.