## 3 Numerical Experiments and Analysis
### The Tuned CC-SHADE-ML
Thus, there is no guarantee that these algorithms will demonstrate the same high performance with other LSGO problems. At the same time, the proposed approach automatically adapts to the given problem, so we conclude that it can also perform well when solving new LSGO problems. In Section 3.3, we propose a hybrid algorithm which is a combination of CC-SHADE-ML and MTS-LS1 [41]. Table 10 contains the detailed results for the fine-tuned CC-SHADE-ML algorithm. The first column contains three checkpoints, 1.2 \(\times\) 10\({}^{5}\), 6.0 \(\times\) 10\({}^{5}\), and 3.0 \(\times\) 10\({}^{6}\). The remaining columns show the number of a benchmark problem. Each cell contains five numbers: the best-found value, the median value, the worst value, the mean value, and the standard deviation value. The authors of the LSGO CEC'2013 benchmark set recommend the inclusion of this information for the convenient further comparison of the proposed algorithm with others. Usually, the comparison is based on values after 3.0 \(\times\) 10\({}^{6}\) FEs. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline
**Number** & **Algorithm** & **Total Win** & **Total Loss** & **Total Equal** & **Averaged Rank** \\ \hline (5) & CC\_tuned-k(7) & 5 & 0 & 70 & 2.50 \\ (3) & CC\_tuned-k(3) & 6 & 2 & 67 & 3.2 \\ (6) & CC\_tuned-k(10) & 2 & 2 & 71 & 3.53 \\ (4) & CC\_tuned-k(5) & 2 & 2 & 2 & 71 & 3.77 \\ (1) & CC\_tuned-k(1) & 0 & 4 & 71 & 4.47 \\ (2) & CC\_tuned-k(2) & 0 & 5 & 70 & 3.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The sum of the Wilcoxon test results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Metaheuristic** & **C1** & **C2** & **C3** & **C4** & **C5** & **Total** \\ \hline SHADEILS & 9.67 & 11.75 & 11.00 & 14.67 & 15.00 & 62.08 \\ MOS & 12.83 & 10.75 & 9.50 & 12.33 & 10.00 & 55.42 \\ MLSHADE-SPA & 13.33 & 12.00 & 11.75 & 12.00 & 3.00 & 52.08 \\ CC-RDG3 & 6.50 & 12.25 & 12.63 & 9.67 & 11.00 & 52.04 \\ BICCA & 11.00 & 10.50 & 11.00 & 9.00 & 8.00 & 49.50 \\ IHDEIS & 9.00 & 8.00 & 8.38 & 12.33 & 9.00 & 46.71 \\ SGCC & 2.00 & 6.63 & 8.38 & 8.33 & 14.00 & 39.33 \\ SACC & 12.17 & 4.75 & 4.50 & 4.67 & 13.00 & 39.08 \\ CC-SHADE-ML & 4.67 & 6.25 & 5.75 & 7.00 & 12.00 & 35.67 \\ CC-CMA-ES & 9.33 & 5.38 & 8.00 & 7.67 & 2.00 & 32.38 \\ VMODE & 6.00 & 5.38 & 6.38 & 7.33 & 6.00 & 31.08 \\ DGSC & 6.67 & 7.38 & 5.38 & 5.00 & 4.00 & 28.42 \\ MPS & 4.33 & 8.25 & 7.50 & 3.00 & 5.00 & 28.08 \\ DECC-G & 10.00 & 6.00 & 4.38 & 3.67 & 1.00 & 25.04 \\ DEEFSO & 2.50 & 4.75 & 5.50 & 3.33 & 7.00 & 23.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The ranking of state-of-the-art metaheuristics. Figure 7: The self-adaptation curves for CC_size and pop_size of CC-tuned-k(7) in an independent run on: (**a**,**c**,**e**,**g**)—CC-size for \(F_{1}\), \(F_{4}\), \(F_{7}\), \(F_{15}\) and (**b**,**d**,**f**,**h**)—pop-size for \(F_{1}\), \(F_{4}\), \(F_{7}\), \(F_{15}\). Figure 8: Convergence plots of CC-tuned-k algorithms. Benchmark problems are: (**a**) \(F_{1}\); (**b**) \(F_{2}\); (**c**) \(F_{3}\); (**d**) \(F_{4^{\prime}}\); (**e**) \(F_{5^{\prime}}\); (**f**) \(F_{6^{\prime}}\); (**g**) \(F_{7^{\prime}}\); (**h**) \(F_{8}\); (**i**) \(F_{9^{\prime}}\); (**j**) \(F_{10^{\prime}}\); (**k**) \(F_{11^{\prime}}\); (**l**) \(F_{12^{\prime}}\); (**m**) \(F_{13^{\prime}}\); (**n**) \(F_{14^{\prime}}\); (**o**) \(F_{15}\).