## Appendix: Numerical tests on the Rosenbrock's function  
To compare the effectiveness of trajectories T1 to T5, the well known [PERSON]'s function has been tested:  
\[f\left(\mathbf{x}\right)=\sum\
olimits_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i }^{2}\right)^{2}+\left(x_{i}-1\right)^{2}\right];\]  
the minimum point is searched in the \(n\)-dimensional box domain \(\left[-2,\,2\right]^{n}\), it is located at \(\left[1\right]^{n}\) and its value is 0. In the tests performed, \(n\)=5 the size of the swarm is \(m\)=50 and the number of iterations is \(s\)=1000. The results of the numerical tests are shown in Tab. 8.  
Trajectory T3 gives much better results than the other trajectories, especially for the best particle. It is interesting to remark the number \(n_{eval}\) of evaluations of the objective function for which the mean \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{Test} & Run’s & Mean & Standard & Best & Worst & \multirow{2}{*}{\(n_{eval}\)} \\  & best & value & deviation & value & value & \\ \hline \multirow{2}{*}{T1} & average & 8.053 & 3.077 & 4.680 & 13.939 & 7580 \\  & particle & 0.009 & 0.001 & 0.007 & 0.011 & 99950 \\  & average & 6.447 & 2.905 & 2.835 & 12.337 & 5990 \\  & particle & 0.007 & 0.003 & 0.003 & 0.013 & 99970 \\  & average & 1.650 & 2.268 & 0.003 & 7.887 & 34130 \\  & particle & 0.002 & 0.005 & 6\(\times 10^{-7}\) & 0.016 & 100000 \\  & average & 2.228 & 1.505 & 0.019 & 4.566 & 41790 \\  & particle & 2.228 & 1.505 & 0.019 & 4.566 & 40110 \\  & average & 1.612 & 1.105 & 0.184 & 3.932 & 39020 \\  & particle & 1.612 & 1.105 & 0.184 & 3.932 & 37600 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of the numerical tests on the Rosenbrock’s function. value of the best average or of the best particle is obtained. For trajectory T3, the mean for the best average is obtained for \(n_{\it eval}\)=34130 while for the best particle \(n_{\it eval}\)=100000, _i.e._, the algorithm still improves the best particle after 1000 iterations. Hence, for this trajectory the gap between \(n_{\it eval}\) of, respectively, the best particle and the best average is \(\Delta n_{\it eval}\)=65870: the trajectory allows for a wide exploration of the search domain, because the best average is obtained after a great number of iterations (341.3 in the mean), which allows to the algorithm a good selection of the swarm and this, in turn, allows for a good improvement of the best particle. Trajectories T4 and T5 show a completely different situation: all the data for the swarm and the particle are identical: the swarm is completely conditioned by the best particle, which drives the convergence of the whole swarm towards its position. The swarm follows the best particle rather quickly, as the values of \(n_{\it eval}\) show: there is a close negative gap between the values of \(n_{\it eval}\) of the best particle and of the best average: \(\Delta n_{\it eval}\)=-1680 for trajectory T4 and \(\Delta n_{\it eval}\)=-1420 for trajectory T5. This means that, for these trajectories, the best particle is obtained always before the best average, and that it closely forces the dynamics of the whole swarm. In addition, \(n_{\it eval}\) in the two cases are not sufficiently large, which means that the algorithm stops too quickly to improve both the best particle and the swarm. The final result is that there is not a sufficient exploration of the search domain and the final values, especially of the best particle, are not as good as for trajectory T3. Trajectories T1 and T2 show another different behaviour: in these cases, \(n_{\it eval}\) is low for the mean of the best average and high for that of the best particle, and \(\Delta n_{\it eval}\) is positive and takes large values, respectively \(\Delta n_{\it eval}\)=92370 and \(\Delta n_{\it eval}\)=93980.