## 3 Proposed ATMALS-GSK Algorithm
### Atmals-Gsk
#### 3.2.6 Overall Operation of ATMALS-GSK
In all simulations on the CEC 2011 benchmark suite, we follow the official recommendation by setting the maximum number of function evaluations (FEs) to 150,000 [55] to ensure a fair comparison with other methods. With a fixed population size of 100, this results in 1500 iterations for each function (since FEs = population size \(\times\) iterations). Each algorithm is run independently 25 times on each function to ensure statistical reliability. All settings for simulations on CEC 2011 have been considered based on the technical report of the CEC 2011 benchmark in [55]. To assess the algorithm's performance on the CEC 2017 benchmarks, solution error is used as the evaluation metric, defined as \(f(x)-f(x^{n})\), where \(x\) is the best solution found in a single run and \(x^{*}\) is the known global optimum. As recommended in [56], errors smaller than \(10^{-8}\) are considered zero. Experiments are conducted across four problem dimensions: \(D=10\), 30, 50, and 100, with FEs set to 10,000 \(\times\)\(D\). Each algorithm is run independently 51 times per function for every dimension. These settings are based on the technical report in [56]. In the following, to provide a thorough evaluation, the experimental results are organized into four main subsections. Section 4.2 focuses on analyzing the performance of ATMALS-GSK on the CEC 2011 and CEC 2017 benchmark functions. Section 4.3 presents a comparison between ATMALS-GSK, the original GSK algorithm, and its other modified variants. In Section 4.4, ATMALS-GSK is benchmarked against several recent metaheuristics. Furthermore, Section 4.5 includes statistical tests using the Wilcoxon signed-rank and Friedman methods to verify the significance of the results. ### Results of ATMALS-GSK

#### 4.2.1 CEC 2011

The comparative analysis between the original GSK algorithm [16] and the proposed ATMALS-GSK algorithm on the CEC 2011 benchmark suite reveals several noteworthy improvements brought about by the proposed enhancements. Table 3 reports the performance results in terms of Best, Median, Worst, Mean, and Standard Deviation (STD) across 25 independent runs, while Table 4 summarizes the statistical significance through the Wilcoxon signed-rank test. \begin{table}
\begin{tabular}{c 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{2-10}  & **Best** & **Median** & **Wost** & **Mean** & **STD** & **Best** & **Median** & **Wost** & **Mean** & **STD** \\ \hline
1 & 0.00 & \(1.85\times 10^{-21}\) & \(1.49\times 10^{1}\) & 3.28 & 5.21



According to the results obtained in Table 3, ATMALS-GSK demonstrates superior robustness and accuracy on a majority of the functions. For instance, in F2, the improvement is particularly noticeable; ATMALS-GSK reached a better minimum of \(-2.40\times 10^{1}\) versus \(-1.35\times 10^{1}\) by GSK, with a wider performance range indicating better exploration capacity. For multimodal functions like F6 and F7, which are known for their local optima traps, ATMALS-GSK consistently outperformed GSK in both mean and worst-case scenarios, showcasing its ability to avoid premature convergence. The mean value for F6 improved significantly from \(-6.94\) to \(-1.79\times 10^{1}\), and for F7, both mean and worst-case values decreased, reflecting more consistent convergence behavior. Notably, in some functions such as F3, F4, and F8, both techniques achieved identical or near-identical performances, indicating that these functions were relatively easy for both methods, leaving little room for improvement. The superiority of ATMALS-GSK becomes even more evident in complex hybrid and composition functions like F17 through F20. These functions typically combine the challenges of high dimensionality, non-separability, and rugged landscapes. On F17, the mean objective value decreased from \(2.09\times 10^{6}\) to \(1.90\times 10^{6}\), and the standard deviation dropped sharply, highlighting more reliable convergence.