## 3 Experiment and Analysis of the Benchmark Functions Test  
### Performance on Uni-Modal and Multi-Modal Benchmark Functions  
The comparison experiments are conducted between the BSLSPIO, PIO, the constriction factor PSO (CFPSO) [48], and the time-varying acceleration coefficients PSO (TVACPSO) [49] over 30 independent runs in Matlab 2017b. The computer configuration is Windows 11 and i5-6300U [EMAIL_ADDRESS] GHz. Thirteen widely used benchmark functions are listed in Table 1, where \(F_{1}\) to \(F_{5}\) are the uni-modal functions, \(F_{6}\) is a step function, and \(F_{7}\) to \(F_{13}\) are the multi-modal functions [50]. The maximum iteration number is 2000. The parameter settings are presented in Table 2. The results of \(N=50\) are listed in Table 3, where the gray shading represents the optimal solution. Figure 4 presents the comparison results of some of the functions. To evaluate the performance of the BSLSPIO, Table 4 provides the testing results of \(N=1000\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{Types} & Name & Range & Optimum \\ \hline \multirow{8}{*}{Uni-modal benchmark functions} & \(F_{1}\) & Sphere model & [\(-\)100,100] \({}^{N}\) & 0 \\  & \(F_{2}\) & [PERSON]’s problem 2.22 & [\(-\)10,10] \({}^{N}\) & 0 \\  & \(F_{3}\) & [PERSON]’s problem 1.2 & [\(-\)10,10] \({}^{N}\) & 0 \\  & \(F_{4}\) & [PERSON]’s problem 2.21 & [\(-\)100,10] \({}^{N}\) & 0 \\  & \(F_{5}\) & Generalized [PERSON]’s functions & [\(-\)30,30] \({}^{N}\) & 0 \\  & \(F_{6}\) & Step function & [\(-\)100,100] \({}^{N}\) & 0 \\  & \(F_{7}\) & Quartic function & [\(-\)128,128] \({}^{N}\) & 0 \\  & \(F_{8}\) & Generalized [PERSON]’s problem 2.26 & [\(-\)500,500] \({}^{N}\) & \(-\)418.9829 \(\times\)\(N\) \\  & \(F_{9}\) & Generalized [PERSON]’s function & [\(-\)512,12] \({}^{N}\) & 0 \\  & \(F_{10}\) & [PERSON]’s function & [\(-\)32,32] \({}^{N}\) & 0 \\  & \(F_{11}\) & Generalized Griewank function & [\(-\)600,600] \({}^{N}\) & 0 \\  & \(F_{12}\) & Generalized Penalized function 1 & [\(-\)50,50] \({}^{N}\) & 0 \\  & \(F_{13}\) & Generalized Penalized function 2 & [\(-\)50,50] \({}^{N}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Uni-modal and multi-modal benchmark functions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline Algorithm & CFPSO & TVACPSO & PIO & BSLSPIO \\ \hline \multirow{8}{*}{Parameter settings} & \(c_{1}=1.49618\) & \(c_{max_{1}}=0.5\) & \(R=0.004\) \\  & \(c_{2}=1.49618\) & \(c_{max_{1}}=2.5\) & \(R=0.004\) & \(N_{p}=100\) \\  & \(c_{2}=1.49618\) & \(c_{min_{2}}=0.5\) & \(N_{p}=100\) & \(N_{S_{p}}=3\) \\  & \(w=0.7298\) & \(c_{max_{2}}=2.5\) & \(T_{max_{1}}=1000\) & \(N_{c}=5\) \\  & \(N_{p}=100\) & \(w_{min_{2}}=0.9\) & \(T_{max_{2}}=1000\) & \(T_{max_{1}}=1000\) \\  & \(w_{max}=0.4\) & & \(T_{max_{2}}=1000\) \\  & & \(N_{p}=100\) & & \(T_{max_{2}}=1000\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings of the compared CFPSO, TVACPSO, PIO, and BSLSPIO. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & _Methods_ & _Arg_ & _Std_ & _Med_ & _Rank_ \\ \hline \multirow{8}{*}{\(F_{1}\)} & CFPSO & \(9.47\times 10^{-27}\) & \(2.97\times 10^{-26}\) & \(9.53\times 10^{-28}\) & 2 \\  & TVPSO & \(2.80\times 10^{-13}\) & \(7.75\times 10^{-13}\) & \(5.33\times 10^{-14}\) & 3 \\  & PIO & \(2136.0847\) & \(1123.6571\) & \(2101.5801\) & 4 \\  & BSLSPIO & 0