## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{6}{c}{**GSK [16]**} & \multicolumn{6}{c}{**ATMALS-GSK (Proposed)**} \\ \cline{2-11}  & **Best** & **Median** & **Worst** & **Mean** & **STD** & **Best** & **Median** & **Worst** & **Mean** & **STD** \\ \hline
1 & \(3.96\times 10^{-1}\) & \(5.38\times 10^{3}\) & \(2.11\times 10^{4}\) & \(5.80\times 10^{3}\) & \(4.63\times 10^{3}\) & \(1.96\times 10^{-1}\) & \(1.98\) & \(7.98\) & \(2.55\) & \(4.79\) \\ \hline
3 & \(6.01\times 10^{4}\) & \(1.15\times 10^{5}\) & \(1.48\times 10^{5}\) & \(1.15\times 10^{5}\) & \(2.15\times 10^{4}\) & \(4.87\times 10^{4}\) & \(4.91\times 10^{4}\) & \(5.30\times 10^{4}\) & \(4.91\times 10^{4}\) & \(1.26\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of GSK and ATMALS-GSK for CEC 2017 benchmark functions in D = 100. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Function**} & \multicolumn{4}{c}{**GSK [16]**} & \multicolumn{4}{c}{**ATMALS-GSK (Proposed)**} \\ \cline{2-11}  & **Best** & **Median** & **Worst** & **Mean** & **STD** & **Best** & **Median** & **Worst** & **Mean** & **STD** \\ \hline
4 & \(8.46\times 10^{1}\) & \(2.17\times



Figure 10: Adaptive behavior of \(K\) in the ATMALS-GSK algorithm vs. iterations in CEC 2017. Figure 11: Adaptive behavior of \(K\), \(K_{F}\), \(K_{R}\), \(P\), and \(PLS\) in the ATMALS-GSK algorithm vs. iterations, on average for all benchmark functions in CEC 2017. Figure 12: _Cont._

Figure 12: Cont. Figure 12: Average convergence curves (over 51 runs) of the original GSK and proposed ATMALS-GSK algorithms for different functions of CEC 2017, considering \(D=10\). ### Comparison with GSK and Modified GSKs

The comparisons of the ATMALS-GSK algorithm with the original GSK and its variants (AGSK, APGSK, FDBAGSK, and eGSK) 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 are provided in Tables 9-12, respectively. As seen in Table 9, for functions 1 to 4 and 6, which demonstrate perfect or near-perfect convergence across all algorithms, the basic GSK algorithm alone is sufficient. For functions 5, 7, and 8, we begin to see notable differences. For example, function 5 showcases that ATMALS-GSK (7.57 \(\pm\) 2.71) falls somewhere between AGSK (6.33 \(\pm\) 1.89) and eGSK (4.82 \(\pm\) 1.90). However, for more rugged terrains like functions 7 and 8, ATMALS-GSK performs more competitively, especially when compared to the original GSK, which suffers from high error means (30.7 and 20.2, respectively). Here, ATMALS-GSK manages to cut down the error significantly (to 16.3 and 7.57), underscoring its adaptive capacity. Functions like 10 and 12 really illustrate the leap ATMALS-GSK offers. In function 10, the original GSK's performance (1060 \(\pm\) 133) is substantially improved by AGSK and APGSK, but ATMALS-GSK achieves a more favorable trade-off (455 \(\pm\) 193), suggesting it maintains robustness even in highly multimodal scenarios. Function 12 reveals an even more dramatic advantage--ATMALS-GSK posts a mean of 11.2 with very low deviation, compared to AGSK's 44.6 and the original's 89.3. Although some modifications, particularly AGSK (0.039 \(\pm\) 0.195), deliver astonishing performance, ATMALS-GSK lags slightly behind (2.78 \(\pm\) 0.93). ATMALS-GSK continues to impress in functions 13, 15, and 16. Especially in function 13, its low mean (2.65) coupled with a reasonable standard deviation suggests more reliable convergence than many competitors. In function 17, however, ATMALS-GSK shows a significant spike in variance (mean = 12.5, SD = 90.2), possibly revealing sensitivity to initial population or parameter randomness in highly stochastic landscapes. This outlier behavior points to a possible need for parameter fine-tuning or incorporating stability controls.