## 6 Results
### Multimodal-Multi-Optima Problems
However, while this results in high precision optima positions, this haziness also simultaneously limits the mobility of numerous particles. This mobility reduction lessens their ability to move toward new optima. On the other hand, larger numbers of particles may successfully compensate for this reduced mobility. Hence, to detect numerous optima concentrated in small zones, such as those of the Shubert and Vincent functions, the procedure must combine a sufficiently large particle number to probe the search space and a high \(\gamma^{*}\) value to form clusters and improve the precision of the identified position locations. However, large particle numbers can rapidly lead to excessive numbers of function evaluations. In clear, in a context of a limited number of function evaluations, too large particle numbers limit the number of successive improvement steps and ultimately reduce the algorithm precision. The best particle number-haziness combination is thus a trade-off. Figure 14: Positions of the 81 optima of the 3D Shubert function (F\({}_{8}\)). \begin{table}
\begin{tabular}{c c c} \hline \hline
**S-EPSO** & **Tie** & **ANDE and NMMSO** \\ \hline
**39** & 59 & 38 \\
**28.7\%** & 43.4\% & 27.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Pairwise comparison of the numbers of best responses. The present analysis does not include any pursuit of the best trade-off. However, to picture the particle number-haziness interrelation, Table 6 shows the results of additional optimizations of the 2D version of the Vincent function F\({}_{7}\). The first case increases \(\gamma^{*}\) to \(2\times 10^{4}\) and maintains the particle number at 1000. The second of these new simulations also increases the particle number to 1500 particles. Again, these choices are arbitrary and only consider convenient values selected larger than those defining the initial setup. As in Tables 4 and 5, the presentation in Table 6 highlights the best performances in boldface type. It also underlines the evaluations improving the performance of the initial setup (\(N\)-1000 and \(\gamma^{*}\)-10\({}^{4}\)). The results of the \(N\)-1000 and \(\gamma^{*}\)-\(2\times 10^{4}\) setup shows that the reduction of the particle mobility brought about by the \(\gamma^{*}\) augmentation first reduces the number of found optima. Indeed, compared to the initial configuration (\(N\)-1000 and \(\gamma^{*}\)-10\({}^{4}\)), \(PR\) and \(SR\) demonstrate smaller values at the first two accuracy levels (\(\epsilon\geq 10^{-2}\)). Conversely, as indicated by the \(PR\) evaluations at higher accuracy levels (\(\epsilon\leq 10^{-3}\)), the \(\gamma^{*}\) augmentation improves the response precision. The second case (\(N\)-1500 and \(\gamma^{*}\)-\(2\times 10^{4}\)) demonstrates that the augmentation of the particle number clearly increases the number of found optima and tends to improve the precision. However, while this \(N\) and \(\gamma^{*}\) configuration is markedly better, at the highest accuracy level (\(\epsilon\leq 10^{-5}\)), its response appears to be a little below that of the two others. Indeed, the search would have benefited from some additional iterations. These observations tend to indicate that it would be possible to identify a more efficient configuration. However, based on a comparison with the results of the two reference algorithms, we may also assume that with this specific landscape type, the best S-EPSO performance would hardly reach those of ANDE and NMMSO. In other words, for landscapes concentrating numerous optima in small zones, strategies such as niching, which somehow force additional links between particles of a cluster, represent an indispensable option. The results obtained for functions F\({}_{11}\) to F\({}_{17}\) correspond to the best S-EPSO performance. In fact, S-EPSO outperforms the two reference algorithms over most of the corresponding evaluations.