# Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems +
## 4 Experimental Setup and Results
### Numerical Experiments
#### 4.2.7 Comparison with Some State-of-The-Art Metaheuristics  
We have compared fine-tuned iCC-SHADE with the following metaheuristics: SHADEILS [51], CC-RDG3 [52], IHDELS [53], SACC [54], MPS [55], VMODE [56], and SGCC [57]. These state-of-the-art metaheuristics were specially created for solving LSGO problems. We have used the Formula-1 scoring system for ranking the mean best-found values. The list of points is the following: \"25, 18, 15, 12, 10, 8, 6, 4, 2, 1\". As we compare eight metaheuristics, the best-performed and the worst-performed algorithm will get 25 and 4 points, respectively.  
Table 15 shows the experimental results Each cell contains the value averaged over 25 independent runs. Table 16 and Figure 14 are based on Table 15 results and the Formula-1 scoring system. In Table 16, the first row denotes the algorithms, the first column denotes classes of functions that are described in Section 4.2.5. Fine-tuned iCC-SHADE has taken fourth place. iCC-SHADE does not dominate in some classes.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Algorithm** & **(2)** & **(3)** & **(4)** & **(5)** \\ \hline (1) & iCC(_pop\_size_-25) & 2/11/2 & 3/11/1 & 3/11/1 & 3/11/1 \\ (2) & iCC(_pop\_size_-50) & - & 3/10/2 & 2/11/2 & 3/9/3 \\ (3) & iCC(_pop\_size_-100) & - & - & 1/3/11 & 2/2/11 \\ (4) & iCC(_pop\_size_-150) & - & - & - & 3/2/10 \\ (5) & iCC(_pop\_size_-200) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 13: The results of the Wilcoxon test for iCC with different population sizes.  
Figure 13: Convergence graphs for iCC on \(F_{1}\) (**a**), and \(F_{7}\) (**b**) with different population size.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & **Algorithm** & **(2)** & **(3)** & **(4)** & **(5)** \\ \hline (1) & iCC(_pop\_size_-25) & 2/11/2 & 3/11/1 & 3/11/1 & 3/11/1 \\ (2) & iCC(_pop\_size_-50) & - & 3/10/2 & 2/11/2 & 3/9/3 \\ (3) & iCC(_pop\_size_-100) & - & - & 1/3/11 & 2/2/11 \\ (4) & iCC(_pop\_size_-150) & - & - & - & 3/2/10 \\ (5) & iCC(_pop\_size_-200) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 13: The results of the Wilcoxon test for iCC with different population sizes.  
The detailed results for fine-tuned iCC-SHADE are presented in Table 17. _Q1_, _Q2_, and _Q3_ denotes 1.2 \(\times\) 10\({}^{5}\), 6.0 \(\times\) 10\({}^{5}\), and 3.0 \(\times\) 10\({}^{6}\) FEVs, respectively. The table contains the best, the median, the worst, the mean, and the standard deviation from 25 independent runs. As we can see, differences between the median and the mean values on all problems are low. The standard deviation values are less than the median and mean values in all experiments, thus the scatter of the results is not high. We can conclude that iCC-SHADE is a rather robust algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\#** & **SHADEILS** & **CC-RDG3** & **IHDELS** & **iCC-SHADE** & **SACC** & **MPS** & **VMODE** & **SGCC** \\ \hline C1 & 30 & 22 & 65 & 33 & 24 & 65 & 17 & 4 \\ C2 & 121 & 160 & 103 & 67 & 73 & 41 & 89 & 64 \\ C3 & 68 & 39 & 45 & 45 & 26 & 12 & 7 & 27 \\ C4 & 18 & 10 & 8 & 6 & 25 & 12 & 1 & 15 \\ Total & 237 & 231 & 221 & 151 & 148 & 130 & 114 & 110 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Scores for fine-tuned iCC-SHADE and state-of-the-art metaheuristics.  
Figure 14: Average ranks of algorithms, comparison of the fine-tuned iCC-SHADE and state-of-the-art metaheuristics.