## 4 Numerical Validation and Performance of Hybrid HCGA Algorithm
### Validation in Numerical Experiments  
In order to verify the performance of the algorithms, cultural algorithm (CA) [16], genetic algorithms (GAs) [11], differential evolution (DE, rand/1/L) [33] and HCGA were selected for comparison with numerical experiments. Ten mathematical functions optimization test problems shown in Table 1 were used to compare the performance of HCGA with GAs, CA and DE. The parameters in HCGA were selected based on the results of the parameter discussion in Section 4.1, and the parameters of each algorithm in the experiments were set as shown in Table 6. Since evolutionary algorithms are essentially stochastic optimization algorithms, the solution found may not be the same each time. Therefore, each benchmark function was repeated 30 times. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{F}\) & \(\  
The optimal values, means and standard deviations of HCGA, GAs, CA and DE for 30 independent runs are listed in Table 7, which were used to evaluate the optimization accuracy, average accuracy and stability of the algorithms. To obtain more reliable statistical conclusions, Wilcoxon nonparametric statistical tests were performed at \(\alpha=0.05\), and the symbols \(+\), \(-\) or \(=\) mean that the optimization results of HCGA were significantly better, worse or similar to the comparison algorithm, respectively. Figure 6 shows the convergence curves of some of the benchmark test functions. The results are summarized as \(+\)/ \(-\)/ \(=\) as the last row of each Table in Table 7.  
As can be seen from Figure 6, HCGA shows a better performance for most functions. For unimodal functions, the convergence speed and accuracy of HCGA are significantly better than those of other algorithms. For multimodal functions, HCGA is able to achieve higher optimization accuracy in shorter iterations for all functions except for function \(F_{10}\), which was slightly inferior to GAs and DE in terms of convergence speed in the initial search stages. This means that HCGA not only has good search ability and fast convergence, but also moderates quite well the conflict well between convergence speed and premature convergence, which means that it has a balanced exploitation and exploration ability. The experimental results in Table 7 show that HCGA performs better for most of the tested functions compared with other algorithms, and can obtain higher optimization accuracy, average accuracy and better stability. This indicates that HCGA is less affected by randomness and that it can maintain optimization accuracy under multiple independent runs. The results of Wilcoxon nonparametric statistical tests for CA, GAs and DEs were 23/0/1, 22/1/1, and 21/3/0, respectively, indicating that the differences between the HCGA and the other three compared algorithms are statistically significant, implying that for all test functions, HCGA shows better performance or is close to the best performance of the other algorithms, which means that it is more robust. In addition, HCGA shows an optimization capability for high-dimensional problems that cannot be matched by CA, GAs and DE. In high-dimensional optimization problems (100 dimensional \(F_{1}-F_{7}\)), HCGA has significant advantages in optimization accuracy, average accuracy and stability. For the functions \(F_{4}-F_{7}\), the number of local optima will increase with the increase in the problem size, and the HCGA does not fall into dimensional disaster; it also scales well with the increasing dimensionality and converges in the proximity of the global optimum, which indicates its high level of performance in solving high-dimensional functions. HCGA can still maintain strong optimization accuracy and robustness in solving high-dimensional optimization problems, which lays the foundation for the application of HCGA in practical problems.