## 4 Analysis of Experimental Results
### Simulation Experiment
#### 4.1.2 Comparative Analysis of Performance with Other Algorithms
Table 5 shows that IGEO has the lowest MAE value and the best performance. In Figure 5, the average convergence curve is also presented in logarithmic form for a better and more precise comparison of the experimental data. The absolute values of all findings are plotted, because \(-1\) is the ideal value of the function F8. In Figure 5, the test results for the convergence speed and search accuracy of the algorithms are presented. The results demonstrate that IGEO outperforms the comparison algorithms on F1 to F8. Specifically, for F1, F2, F3, F6, and F7, IGEO shows significant advantages in terms of average value, indicating an overwhelmingly high convergence speed and search accuracy. For F4, F5, and F8, IGEO exhibits better performance than the comparison algorithms, and the ability to accurately capture the theoretical optimal values of 0.9 and \(-1\) for F5 and F8, respectively. While IGEO shows inferior performance on F9, F10, F11, and F12, its standard deviation is 0, demonstrating superior robustness compared to other algorithms. Regarding F9, IGEO's performance is second only to WOA. The convergence speed of GWO and the search accuracy of WOA on F9 are comparable to those of IGEO. For F10, GWO outperforms both IGEO and WOA in terms of performance. For F11 and F12, although the average convergence curves of PSO and IGEO are similar, the results presented in Table 4 clearly indicate that IGEO exhibited a higher search accuracy than PSO. Table 4 and Figure 5 present the optimization results for the benchmark functions F1 to F8 in general dimensions (dim = 30), where it is evident that IGEO achieved the best results in terms of both convergence speed and search accuracy. To verify the comprehensive performance and robustness of IGEO, experiments were conducted on the eight benchmark functions under 100-dimensional conditions using the comparison algorithms. The conditional parameters of each algorithm were consistent with those shown in Table 3, and each algorithm was independently run 50 times for each function. Table 6 presents the results of the comparison with the other seven original algorithms, including mean and standard deviation (std). Table 6 clearly demonstrates that IGEO outperformed the other algorithms in terms of both average value and standard deviation, indicating its superior exploration and exploitation ability. Here, the optimization outcomes for the CEC2021 test set are also displayed to further demonstrate IGEO's efficacy. Recent winners are contrasted with IGEO. Both traditional improvements to particle swarm optimization, such EIW_PSO [54], and advances in new swarm intelligence algorithms, like FMMPA [15], are included among the comparison algorithms. Additionally, IGEO is contrasted with upgraded GEO algorithms, including GEO_DLS [42], CMA-ES-RIS [55], and L-SHADE [56]. Table 7 displays the specific outcomes. The data in the table were derived by deducting these values, because none of the 10 functions in the CEC2021 test set have an ideal value of 0. The table includes the minimum value, maximum value, standard deviation, and average value of the algorithm \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **MAE** & **Ranking** & **Algorithm** & **MAE** & **Ranking** \\ \hline IGEO & \(9.25\times 10^{-17}\) & 1 & GEO & \(4.93\times 10^{-1}\) & 6 \\ WOA & \(8.53\times 10^{-2}\) & 2 & GWO & \(5.42\times 10^{-1}\) & 7 \\ GEO\_DLS & \(1.61\times 10^{-1}\) & 3 & SSA & \(1.50\times 10^{0}\) & 8 \\ PSO & \(3.62\times 10^{-1}\) & 4 & BOA & \(2.59\times 10^{0}\) & 9 \\ SCA & \(4.09\times 10^{-1}\) & 5 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of MAE ranking between IGEO and other advanced algorithms. optimization. The table shows that IGEO's performance is still fairly strong, overall, despite the fact that it is not the best for every function. Figure 5: IGEO and other algorithms’ convergence curves on the test functions.