## 4 Simulation Results and Comparisons  
This section presents the results for the 15 global numerical optimization problems in Table 4, which were used for the performance evaluations of the proposed AUFPSO-NTE algorithm. In Example (1), functions \(f_{1}\)-\(f_{7}\) were used to compare the performance of the proposed AUFPSO-NTE, the PSO-FOV (PSO with the fractional-order velocity) proposed by [PERSON] et al. [34], the FPSO improved by [PERSON] et al. [35] from PSO-FOV, the modified PSO (MPSO) proposed by [PERSON] and [PERSON] [11], and the PSO proposed by [PERSON] and [PERSON] [1]. In Example (2), functions \(f_{5}\) and \(f_{8}\)-\(f_{11}\) were used to compare the performance of the proposed AUFPSO-NTE, the FPSO-based algorithms proposed by [PERSON] et al. [36], and the PSO proposed by [PERSON] et al. [36]. In Example (3), functions \(f_{5}\), \(f_{8}\), \(f_{10}\), \(f_{11}\), and \(f_{12}\) were used to compare between the AUFPSO-NTE, the adaptive fractional-order Darwinian PSO (AFO-DPSO) proposed by [PERSON] et al. [38], and the PSO-based algorithms described in [PERSON] et al. [38]. In Example (4), functions \(f_{5}\), \(f_{8}\), \(f_{10}\), \(f_{11}\), and \(f_{13}\)-\(f_{15}\) were used to compare between the AUFPSO-NTE, the fractional-order Darwinian PSO (FDPSO) proposed by [PERSON] et al. [39] and the PSO-based algorithms described in [PERSON] et al. [39]. The simulations were run on a Windows 10 personal computer with a core i7-6700M, a 3.4 GHz CPU, and 8 GB RAM. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Definition** & **Solution Space** & **Optimal Value** \\ \hline Bohachevsky 1 & \(f_{1}=v_{1}^{2}+2v_{2}^{2}-0.3\cos(3\pi v_{1})-0.4\cos(4\pi v_{2})+0.7\) & \([-50,50]^{2}\) & 0 \\ \hline \multirow{4}{*}{Colville} & \(f_{2}=100\Big{(}v_{2}-v_{2}^{2}\Big{)}^{2}+(1-v_{1})^{2}+90\Big{(}v_{4}-v_{3} ^{2}\Big{)}^{2}+(1-v_{3})^{2}\) & \multirow{4}{*}{\(0\)} \\  & \(+10.1\Big{(}v_{2}-1)^{2}+(v_{4}-1)^{2}\Big{]}\) & \([-10,10]^{4}\) & 0 \\  & \(+19.8\Big{(}v_{2}-1)(v_{4}-1)\) & & \\ \hline \multirow{2}{*}{Drop wave} & \(f_{3}=-\frac{1+\cos\left(\frac{12}{v_{1}^{2}+v_{2}^{2}}\right)}{0.5\Big{(}v_{ 1}^{2}+v_{2}^{2}\Big{)}+2}\) & \([-10,10]^{2}\) & \(-1\) \\ \hline \multirow{2}{*}{Easom} & \(f_{4}=-\cos(v_{1})\cos(v_{2})e^{-(v_{1}-v_{2})^{2}-(v_{2}-v_{3})^{2}}\) & \([-100,100]^{2}\) & \(-1\) \\ \hline \multirow{2}{*}{Rastrigin} & \(f_{5}=\sum_{i=1}^{C}\left(v_{i}^{2}-10\cos(2\pi v_{i})+10\right)\) & \([-5.12,5.12]^{C}\) & 0 \\ \hline \multirow{2}{*}{[PERSON]} & \(f_{6}=\sum_{i=1}^{C}-\sin(v_{i})\left[\sin\frac{(v_{i}+v_{2}^{2})}{\pi}\right] ^{2}\) & \([0,\pi]^{2}\) & \(-1.8409\) \\ \hline \multirow{2}{*}{Rosenbrock’s valley} & \(f_{7}=\sum_{i=1}^{C}100\Big{(}v_{i+1}-v_{i}^{2}\Big{)}^{2}\) & \([-2.048,2.048]^{2}\) & 0 \\ \hline \multirow{2}{*}{Sphere} & \(f_{8}=\sum_{i=1}^{C}v_{i}^{2}\) & \([-100,100]^{C}\) & 0 \\ \hline \multirow{2}{*}{Ackley} & \(f_{9}=-20\exp\left(-0.2\sqrt{\frac{v_{1}^{2}+v_{2}^{2}}{C}}\right)+\exp\left( \sqrt{\frac{v_{1}^{2}+\cos 2\pi v_{1}}{C}}\right)+20\) & \([-32,32]^{C}\) & 0 \\ \hline \multirow{2}{*}{Rosenbrock} & \(f_{10}=\sum_{i=1}^{C-1}\left[100\Big{(}v_{i+1}-v_{i}^{2}\Big{)}^{2}+(1-v_{i})^{2}\right]\) & \([-30,30]^{C}\) & 0 \\ \hline \multirow{2}{*}{[PERSON]} & \(f_{11}=\frac{1}{400}\sum_{i=1}^{C}v_{i}^{2}-\Pi_{i=1}^{C}\cos\left(\frac{v_{i}} {v_{i}}\right)+1\) & \([-600,600]^{C}\) & 0 \\ \hline \multirow{2}{*}{Dejong F4} & \(f_{12}=\sum_{i=1}^{C}v_{i}^{4}\) & \([-20,20]^{C}\) & 0 \\ \hline \multirow{2}{*}{Schwefel’s P1.2} & \(f_{33}=\sum_{i=1}^{C}\left(\sum_{i=1}^{C}v_{i}\right)^{2}\) & \([-100,100]^{C}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Benchmark functions.