## 3 Numerical Experiments and Analysis
### CC-SHADE-ML  
We evaluated the performance of CC-SHADE-ML and compared it with CC-SHADE with a fixed number of subcomponents and a fixed number of individuals. The proposed CC-SHADE-ML algorithm has the following parameters. The set of subcomponents is equal to \(\{1,\,2,\,5,\,10,\,20,\,50,\,100,\,200,\,500,\,1000\}\). The set of the population size is equal to \(\{25,\,50,\,100,\,150,\,200\}\). The number of cycles is set to 50. According to our numerical experiments, this value for the number of cycles performs better than other tested values. Thus, in each cycle, CC-SHADE-ML evaluates \(6.0\,\times\,10^{4}\) FEs. We use \"CC\" and \"CC-k(v)\" notations to save space in Table 2, where CC is all variants of CC-SHADE parameters, and CC-k(v) is the proposed approach, where the parameter \(k\) is set to \(v\) (3), and \(v\) is the power of the exponent in Equation (3). We investigated \(v\) equal to \(\{1,\,2,\,3,\,5,\,7,\,10\}\). Table 2 proves the statistical difference in the results of the rank comparison using the Wilcoxon rank-sum test with the \(p\)-value equal to 0.05. The first column denotes better (+), worse (-), and equal performance (\(\approx\)). Other columns contain the settings of the proposed algorithm. The cells contain the total number of benchmark problems where CC demonstrates better, worse, or equal performance in comparison with CC-k(v). As we can see, each version of the proposed algorithm has scores larger than CC. As we can see from Table 2, the proposed algorithm with all values of the power (3) always demonstrates better performance in comparison with the CC with a fixed number of subcomponents and individuals. Based on the results of the numerical experiments and the results of the statistical test, it is preferable on average to choose the proposed approach than the CC algorithm with an arbitrary set of parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Benchmark Problem** & **The Best Combination(s)** & **Class** \\ \hline
1 & \(20\,\times\,50\) & C1 \\
2 & \(1000\,\times\,200\) & C1 \\
3 & \(1000\,\times\,50,1000\,\times\,100,1000\,\times\,150,\,1000\,\times\,200\) & C1 \\
4 & \(1\,\times\,200\) & C2 \\
5 & \(1\,\times\,200\) & C2 \\
6 & \(1000\,\times\,200\) & C2 \\
7 & \(5\,\times\,200\) & C2 \\
8 & \(1\,\times\,200\) & C3 \\
9 & \(5\,\times\,150\) & C3 \\
10 & \(1000\,\times\,200\) & C3 \\
11 & \(5\,\times\,200\) & C3 \\
12 & \(50\,\times\,25\) & C4 \\
13 & \(5\,\times\,150\) & C4 \\
14 & \(2\,\times\,200\) & C4 \\
15 & \(200\,\times\,25\) & C5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The best combination(s) of parameters for CC-SHADE on LSGO CEC’2013. Figure 2: The ranks sum of CC-SHADE algorithms’ parameters for all benchmark problems. We have compared the performance estimations of all CC-k algorithms. The statistical experimental results are placed in Table 3. The first column denotes indexes of the algorithm. The second column denotes the title of the algorithm. The next columns denote the compared algorithms corresponding to the index value. Values in each cell are based on the following notation. We compare algorithms from a row and column, if the algorithm from the row demonstrates statistically better, worse of equal performance we add points to the corresponding criterion. Table 3 contains the sum of (_better/t worse/equal_) points of all algorithms. The results in Table 4 are based on the results from Table 3. Algorithms are sorted according to their statistical performance and their averaged rank. As we can see, CC-k(7) has taken first place. It outperforms the other algorithms 10 times, loses only once and demonstrates the same performance 64 times. It can be noted that the second last column contains large values. This means that the majority of considered algorithms demonstrate an equal performance on benchmark problems that can be explained by the introduced self-adaptiveness. Based on the ranking and the statistical tests, we can conclude that CC-k(7) performs the other variants of CC-k(v).