## 4 Validation of the Proposed Method
### INPSO Algorithm for Numerical Function Optimization
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function Expression** & **Name** & \(\mathbf{d}\) & **Search Interval** & \(\mathbf{f_{\min}}\) \\ \hline \(F_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & Sphere & 30 & \([-100,100]^{n}\) & 0 \\ \(F_{2}(x)=\sum\limits_{i=1}^{n}\left|x_{i}^{2}\right|+\prod\limits_{i=1}^{n} \left|x_{i}\right|\) & Schwefel 2.22 & 30 & \([-10,10]^{n}\) & 0 \\ \(F_{3}(x)=\sum\limits_{i=1}^{n}\left|\left\langle\sum\limits_{j=1}^{n}x_{j} \right\rangle^{2}\) & Schwefel 1.2 & 30 & \([-100,100]^{n}\) & 0 \\ \(F_{4}(x)=\max_{i}(x_{i}\left|x_{i}\right|,1\leq i\leq n)\) & Schwefel 2.21 & 30 & \([-100,100]^{n}\) & 0 \\ \(F_{5}(x)=\sum\limits_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+ \left(x_{i}-1\right)^{2}\right]\) & Rosenbrock & 30 & \([-30,30]^{n}\) & 0 \\ \(F_{6}(x)=\sum\limits_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & Rastrigin & 30 & \([-5.12,5.12]^{n}\) & 0 \\ \(F_{7}(x)=\frac{1}{1000}\sum\limits_{i=1}^{n}x_{i}^{2}-\prod\limits_{i=1}^{n} \cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & Griewank & 30 & \([-50,50]^{n}\) & 0 \\ \(F_{8}(x)=\frac{n}{n}\left|10\sin(\pi y_{1})\right.\) & & & & \\ \(+\sum\limits_{i=1}^{n}\left(y_{i}-1\right)^{2}\left[1+10\sin^{2}(\pi y_{i}+1)\right]\) & Penalized & 30 & \([-50,50]^{n}\) & 0 \\ \(+(y_{n}-1)^{2}\right]+\sum\limits_{i=1}^{n}u(x_{i},10,100,4)\) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Benchmark functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **Indicators** & \multicolumn{4}{c}{**Algorithms**} \\ \cline{3-6}  & **SCA** & **CS** & **PSO** & **INPSO** \\ \hline \multirow{3}{*}{\(F_{1}\)} & Mean & 38.9414 & 11.5363 & \(1.26\times 10^{-4}\) & \(\mathbf{5.97\times 10^{-63}}\) \\  & S.D. & 72.5009 & 3.9837 & 0.0002 & \(\mathbf{3.27\times 10^{-62}}\) \\  & Best & 0.0866 & 5.7200 & \(5.61\times 10^{-6}\) & \(\mathbf{1.49\times 10^{-84}}\) \\ \hline \multirow{3}{*}{\(F_{2}\)} & Mean & 0.0364 & 5.9192 & 7.0456 & \(\mathbf{1.34\times 10^{-54}}\) \\  & S.D. & 0.0508 & 1.9614 & 9.5127 & \(\mathbf{6.91\times 10^{-54}}\) \\  & Best & 0.0012 & 3.2551 & 0.0084 & \(\mathbf{5.94\times 10^{-67}}\) \\ \hline \multirow{3}{*}{\(F_{3}\)} & Mean & 8023.4889 & 2475.35 & 82.4309 & \(\mathbf{0.0010}\) \\  & S.D. & 4265.1060 & 607.633 & 40.6034 & \(\mathbf{0.0038}\) \\  & Best & 867.6201 & 1310.4708 & 91.9355 & \(\mathbf{2.58\times 10^{-6}}\) \\ \hline \multirow{3}{*}{\(F_{4}\)} & Mean & 36.2101 & 11.1260 & 1.0780 & \(\mathbf{7.63\times 10^{-9}}\) \\  & S.D. & 10.7971 & 1.7631 & 0.2270 & \(\mathbf{4.12\times 10^{-8}}\) \\  & Best & 17.3797 & 7.7900 & 0.5766 & \(\mathbf{3.04\times 10^{-15}}\) \\ \hline \multirow{3}{*}{\(F_{5}\)} & Mean & 30779.15 & 623.5256 & 177.5357 & \(\mathbf{26.5895}\) \\  & S.D. & 44845.7 & 354.2101 & 551.9438 & \(\mathbf{0.1765}\) \\  & Best & \(\mathbf{17.3797}\) & 242.6105 & 21.6433 & \(26.1215\) \\ \hline \multirow{3}{*}{\(F_{6}\)} & Mean & 36.2101 & 100.6785 & 102.8620 & \(\mathbf{13.4779}\) \\  & S.D. & 38.1728 & \(\mathbf{14.0177}\) & 27.3967 & \(24.7228\) \\ \cline{1-1}  & Best & 0.0088 & 77.2391 & 47.6199 & \(\mathbf{0}\) \\ \hline \multirow{3}{*}{\(F_{7}\)} & Mean & 0.9152 & 1.1088 & 0.0080 & \(\mathbf{0.0041}\) \\  & S.D. & 0.3664 & 0.0389 & \(\mathbf{0.0088}\) & \(0.0226\) \\ \cline{1-1}  & Best & 0.0971 & 1.0470 & \(8.27\times 10^{-7}\) & \(\mathbf{0}\) \\ \hline \multirow{3}{*}{\(F_{8}\)} & Mean & 100,099 & 3.8871 & 0.0622 & \(\mathbf{0.0231}\) \\ \cline{1-1}  & S.D. & 417,103 & 0.8850 & 0.0843 & \(\mathbf{0.0818}\) \\ \cline{1-1}  & Best & 0.0689 & 1.9252 & \(8.91\times 10^{-7}\) & \(\mathbf{3.44\times 10^{-7}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimization results on unimodal and multimodal functions. Figure 4: Convergence trends for INPSO versus other optimizers: (**a**) Sphere function, (**b**) Schwefel 2.22 function, (**c**) Schwefel 1.2 function, (**d**) Schwefel 2.21 function, (**e**) Rosenbrock function, (**f**) Rastrigin function, (**g**) Griewank function, (**h**) penalized function.