## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
Similar improvements can be seen across F18 to F20, where ATMALS-GSK consistently provided better mean performance and significantly lower standard deviations. The Wilcoxon signed-rank test in Table 4 confirms the overall effectiveness of ATMALS-GSK. With a P-value of \(0.0168\), the improvement is statistically significant at the \(5\%\) level. The test outcomes include \(12\) functions where ATMALS-GSK outperformed GSK, \(5\) where the two algorithms tied, and only \(5\) cases where GSK was better. The positive ranking sum (R+ = \(127\)) far exceeds the negative sum (R\(-=26\)), clearly supporting the conclusion that ATMALS-GSK provides a significant and consistent performance gain over the original GSK across diverse benchmark functions. The detailed analysis confirms that ATMALS-GSK is a promising enhancement over the baseline GSK. As mentioned earlier, the proposed ATMALS-GSK algorithm incorporates adaptive parameters. Figure 7 shows the adaptive value of \(K\) across iterations for functions \(1\)-\(6\) as examples, along with the average value over all functions. Figure 8 illustrates the variations of the parameters \(K\), \(K_{F}\), \(K_{R}\), \(P\), and \(PLS\) during the algorithm's execution. Additionally, Figure 9 presents the average convergence curves (over \(25\) runs) for the GSK and ATMALS-GSK algorithms on various functions from the CEC \(2011\) benchmark. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithms** & **R+** & **R\(-\)** & \(\mathbf{p}\)**Value** & **+** & **=** & **Dec.** \\ \hline ATMALS\(-\)GSK vs. GSK & 127 & 26 & 0.0168 & 12 & 5 & 5 & + \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Wilcoxon signed-rank test of ATMALS-GSK along with the original GSK for CEC 2011. Figure 7: Adaptive behavior of \(K\) in the ATMALS-GSK algorithm vs. iterations in CEC 2011. Figure 8: Adaptive behavior of \(K\), \(K_{F}\), \(K_{R}\), \(P\), and \(P_{LS}\) in the ATMALS-GSK algorithm vs. iterations, on average for all benchmark functions in CEC 2011. Figure 9: _Cont._

Figure 9: Average convergence curves (over 25 runs) of the GSK and proposed ATMALS-GSK algorithms for different functions of CEC 2011. #### 4.2.2 Cec 2017

The experimental results of the original GSK algorithm [16] and the proposed ATMALS-GSK algorithm over 51 independent runs are provided in Tables 5-8 for \(D=10\), 30, 50, and 100, respectively. The results obtained highlight the improvements brought by ATMALS-GSK over GSK, particularly in terms of robustness, accuracy, and stability across most benchmark functions in the 10-dimensional test setting. It can be seen in Table 5 that both algorithms perform identically on some simpler or unimodal functions such as F1, F3, F4, F6, and F9. In these cases, all metrics are consistently zero, indicating that both algorithms successfully find the global optimum with perfect reliability. This suggests that the difficulty level of these functions is relatively low for both approaches. However, when it comes to more complex and multimodal functions, ATMALS-GSK demonstrates a significant edge. For instance, in F5, the mean fitness value achieved by ATMALS-GSK is 7.57, substantially lower than the 20.3 reported for GSK, with a reduced standard deviation. This trend is consistently observed in F7 and F8, where ATMALS-GSK yields much better best and mean results, indicating not only better convergence but also enhanced robustness against local optima. The improvements are even more evident in challenging composite and hybrid functions. Take F10 as an example: while GSK results in a mean error of 1060, ATMALS-GSK drastically reduces this to 455, cutting the error by more than half. A similar pattern is seen in F12, F13, and F14, where ATMALS-GSK not only achieves better average fitness values but also shows much lower variability (i.e., lower standard deviation), signifying higher consistency across multiple runs.