## 3 Numerical Experiments and Analysis
### The Tuned CC-SHADE-ML  
In this subsection, we evaluate the performance of the tuned CC-SHADE-ML. As we can see in Figure 2, the region with the best-ranked solutions covers the set of subcomponents equal to \(\{5,\ 10,\ 20,\ 50\}\) and the set of the population size equal to \(\{25,\ 50,\ 100\}\). The tuned CC-SHADE-ML will use these parameters. Figure 6 has the same structure as Figure 3. Based on the ranking, CC_tuned-k(1) demonstrates the best performance. Table 5 has the same structure as Table 2. Table 2 compares the performance of CC and CC_tuned-k algorithms. According to the results of the Wilcoxon test, we can conclude that any power value of tuned CC-k demonstrates better performance than CC with the fixed number of subcomponents and individuals. Tables 6 and 7 have the same structure as Tables 3 and 4, respectively. We placed CC_tuned-k(7) on the first place because it has no loss point. Although, if we take into account only the averaged rank, we then need to place CC_tuned-k(7) in last place. Additionally, CC_tuned-k(1) has the highest rank, however, it does not significantly outperform any of compared algorithms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Algorithm** & **(2)** & **(3)** & **(4)** & **(5)** & **(6)** \\ \hline (1) & CC\_tuned-k(1) & 0/0/15 & 0/1/14 & 0/1/14 & 0/1/14 & 0/1/14 \\ (2) & CC\_tuned-k(2) & - & 0/2/13 & 0/0/15 & 0/2/13 & 0/1/14 \\ (3) & CC\_tuned-k(3) & - & - & 2/1/12 & 0/1/14 & 1/0/14 \\ (4) & CC\_tuned-k(5) & - & - & - & 0/0/15 & 0/0/15 \\ (5) & CC\_tuned-k(7) & - & - & - & - & 1/0/14 \\ (6) & CC\_tuned- & - & - & - & - & - \\ \end{tabular}
\end{table} 
Table 6: The results of the Wilcoxon test for CC-k with different parameter values. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**CC vs.** & **CC\_tuned-k(1)** & **CC\_tuned-k(2)** & **CC\_tuned-k(3)** & **CC\_tuned-k(5)** & **CC\_tuned-k(7)** & **CC\_tuned-k(10)** \\ \hline + & 59 & 57 & 52 & 52 & 57 & 51 \\ - & 84 & 86 & 91 & 90 & 86 & 90 \\ \(\approx\) & 37 & 37 & 37 & 38 & 37 & 39 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The Wilcox rank-sum test CC vs. CC_tuned-k.  
Figure 6: The ranking of CC-tuned-k algorithms. We compared the performance of CC-k(7) and CC_tuned-k between each other using the Wilcoxon test. The statistical difference analysis is presented in Table 8. Columns denote the number of benchmark problems. The (+/-/\(\approx\)) symbols mean better, worse, and equal performance of CC-k in comparison with CC_tuned-k. Algorithms demonstrate the same performance for six problems. CC-k outperforms CC_tuned-k on four problems and loses on five problems. Figure 7 has the same structure as Figure 4. As we can see, on the \(F_{1}\) benchmark problem, the algorithm has chosen a good combination of parameters and does not change the certain number of cycles because this combination demonstrates the high performance. On other benchmark problems, we can see more rapid switching between values of parameters. Figure 8 has the same structure as Figure 5. It shows convergence graphs for CC_tuned algorithms. We have compared the performance of CC_tuned-k(7) with other state-of-the-art LSGO metaheuristics. These metaheuristics were specially created and tuned to solve the LSGO CEC'2013 benchmark set. We selected high-performed metaheuristics from the TACOLab database [26]: SHADEILS [27], MOS [28], MLSHADE-SPA [29], CC-RDG3 [30], BICCA [31], IHDELS [32], SGCC [33], SACC [34], CC-CMA-ES [35], VMODE [36], DGSC [37], MPS [38], DECC-G [39], and DEEPSO [40]. Figure 9 shows the ranking of the compared metaheuristics. Table 9 consists of the ranking values of state-of-the-art algorithms depending on the class of benchmark problems. Ranks are averaged in each class. The proposed algorithm takes ninth place out of 15. We should note that the majority of metaheuristics in the comparison use special local search techniques adapted for the CEC'13 LSGO benchmark and their control parameters are also fine-tuned to the given problem set.