## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
** & **GSK**[16] & **AGSK**[46] & **APCSK**[45] & **FDBAGSK**[36] & **eGSK**[49] & **ATMALS-GSK** \\ \hline
1 & \(5.80\times 10^{2}\pm 4.63\times 10^{3}\) & \(5.08\times 10^{2}\pm 1.19\times 10^{2}\) & \(4.38\times 10^{2}\pm 2.89\times 10^{3}\) & \(5.99\pm 9.79\) & \(9.58\times 10^{2}\pm 4.05\times 10^{4}\) & \(2.55\pm 4.79\) \\ \hline
3 & \(1.15\times 10^{2}\pm 2.15\times 10^{4}\) & \(1.08\times 10^{3}\pm 2.11\times 10^{4}\) & \(7.83\times 10^{4}\pm 9.64\times 10^{3}\) & \(5.55\times 10^{4}\pm 1.42\times 10^{4}\) & \(2.00\times 10^{3}\pm 7.54\times 10^{3}\) & \(4.91\times 10^{4}\pm 1.26\times 10^{4}\) \\ \hline
4 & \(2.65\times 10^{2}\pm 4.57\times 10^{3}\) & \(1.68\times 10^{2}\pm 4.47\times 10^{3}\) & \(2.10\times 10^{2}\pm 5.52\times 10^{3}\) & \(2.07\times 10^{2}\pm 4.53\times 10^{3}\) & \(6.30\pm 1.87\times 10^{3}\) & \(4.90\pm 1.45\times 10^{3}\) \\ \hline
5 & \(5.51\times 10^{2}\pm 3.39\times 10^{2}\) & \(7.28\times 10^{2}\pm 5.28\times 10^{3}\) & \(7.72\times 10^{2}\pm 4.46\times 10^{3}\) & \(6.66\times 10^{2}\pm 5.57\times 10^{3}\) & \(1.12\times 10^{2}\pm 1.72\times 10^{3}\) & \(9.70\times 10^{2}\pm 2.17\times 10^{3}\) \\ \hline
6 & \(1.72\times 10^{-3}\pm 6.46\times 10^{-3}\) & \(2.01\times 10^{-2}\pm 1.20\times 10^{-2}\) & \(3.30\times 10^{-1}\pm 2.97\times 10^{-1}\) & \(1.63\times 10^{-2}\pm 1.27\times 10^{-2}\) & \(4.01\times 10^{-2}\pm 5.90\times 10^{-2}\) & \(1.24\times 10^{-2}\pm 2.73\times 10^{-2}\) \\ \hline
7 & \(8.75\times 10^{2}\pm 1.83\times 10^{3}\) & \(8.73\times 10^{2}\pm 5.20\times 10^{3}\) & \(1.17\times 10^{3}\pm 6.49\times 10^{3}\) & \(8.03\times 10^{2}\pm 5.29\times 10^{3}\) & \(2.13\times 10^{2}\pm 2.45\times 10^{3}\) & \(1.68\times 10^{2}\pm 1.91\times 10^{3}\) \\ \hline
8 & \(4.92\times 10^{2}\pm 3.41\times 10^{2}\) & \(7.15\times 10^{2}\pm 4.81\times 10^{3}\) & \(7.82\times 10^{2}\pm 4.48\times 10^{3}\) & \(6.47\times 10^{2}\pm 5.60\times 10^{3}\) & \(1.12\times 10^{2}\pm 1.66\times 10^{3}\) & \(9.65\times 10^{4}\pm 1.12\times 10^{3}\) \\



### Comparison with Other Metaheuristic Algorithms

In this section, the performance of ATMALS-GSK is compared with recent metaheuristics (i.e., IADE, GJO-JOS, QCSCA, cSM, and FOWFO) in terms of the average and standard deviation results (Mean \(\pm\) STD) over 51 runs for different dimensions \(D\) = 10, 30, 50, and 100 on CEC 2017, as reported in Tables 13-16, respectively. Table 13 highlights the strong performance of the ATMALS-GSK algorithm against other metaheuristics across a range of 10-dimensional benchmark functions. It consistently outperforms or matches competing methods, showing exceptional accuracy and robustness--often achieving exact solutions (e.g., 0.00 with zero variance) for simpler functions like F1, F3, and F4, while remaining highly competitive on more complex or multimodal ones like F5 and F10. Even on the most challenging hybrid and composite functions (F14-F29), ATMALS-GSK frequently delivers near-optimal results with low variability, outperforming methods like IADE, GJO-JOS, and QCSCA in many cases. Its ability to maintain low standard deviations throughout confirms its reliability and stable search behavior, making it a dependable choice for diverse and complex optimization tasks. The results in Tables 13-16 for different dimensions demonstrate the superior performance of ATMALS-GSK for solving high-dimensional problems. Its hybrid mechanism--blending adaptive learning and guided search via GSK--allows it to effectively balance exploration and exploitation. This advantage is evident not only in reaching optimal or near-optimal solutions but also in doing so reliably across a broad set of benchmark problems, outperforming recent metaheuristic algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fun. ** & **IADE**[50] & **GJO-JOS**[51] & **QCSCA**[52] & **cSM**[53] & **FOWPD [54]** & **ATMALS-GSK** \\ \hline
1 & \(8.54\times 10^{2}\pm 1.51\times 10^{3}\) & \(2.30\times 10^{4}\pm 1.40\times 10