## 4 Numerical Validation and Performance of Hybrid HCGA Algorithm  
### Parameter Discussion  
Tuning parameters properly is very important for an evolutionary algorithm to achieve good performance. In HCGA, there are seven main parameters: \(P_{c}\), \(P_{m}\), \(\alpha\), \(\gamma\), \(\mu\), \(K\), \(p\). In this section, we used the factorial design (FD) [16] approach in order to obtain a guideline on how to tune the designed parameters in HCGA. Ten benchmark mathematical optimization problems were used to evaluate and compare optimization algorithms. These functions can be divided into unimodal functions and multimodal functions. Functions \(F_{1}-F_{4}\) are unimodal functions with only a global optimal value, which are mainly used to evaluate the exploitation ability and convergence speed of the algorithm. Functions \(F_{5}-F_{10}\) are multimodal functions, which have multiple local optimal values in the search space, and the number of local optima will increase with the increase of the problem size, which is an important reference for assessing the exploration capability of the algorithm. Seven of these test functions \((F_{1}-F_{7})\) are dimension-wise scalable. The details of the test functions are listed in Table 1.  
In the experiments, the population size was set to twice the dimension for the \(F_{1}-F_{7}\) function and five times the dimension for the \(F_{8}-F_{10}\) function, and the \(T_{max}\) was set to 5000. As shown in Table 2, we used seven parameters as factors for seven levels in an orthogonal experimental design. Table 3 shows the test results of the orthogonal parameter table with the \(10D-f_{1}\) function. Trials of 30 times were performed for each set of parameters. The unabridged result tables, similarly to Table 3 of other experiments, were too large, and they were omitted here. As shown in Table 3, to estimate the effects of each set of parameters, the mean fitness of the 30 runs were obtained and listed in in the last column of the table. \(K_{i}\) is the mean value of mean fitness for this column parameter at level \(i\) (\(i=1,2,\ldots,7\)). Std is the standard deviation of each column \(K_{1}-K_{7}\). The larger the Std value is, the more this column parameter influences the algorithm performance. Furthermore, for each column, if the value of \(K_{i}\) is the smallest \(K\) value in that column, then the best value of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test Function** & **D** & **Search Range** & **Optimal** \\ \hline \(F_{1}=\sum\limits_{i=1}^{n}x_{i}^{2}\) & \(10/30/100\) & \((-100,100)^{D}\) & 0 \\ \(F_{2}=\sum\limits_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|\) & \(10/30/100\) & \((-10,10)^{D}\) & 0 \\ \(F_{3}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{K}x_{j}\) & \(10/30/100\) & \((-100,100)^{D}\) & 0 \\ \(F_{4}=\sum\limits_{i=1}^{n}[(x_{4i-3}+10x_{4i-2})^{2}+5(x_{4i-1}-x_{4i})^{2}+ (x_{4i-2}-\) & \(12/32/100\) & \((-4,5)^{D}\) & 0 \\ \(2x_{4i-1})^{4}+10(x_{4i-3}-x_{4i})^{4}]\) & & & \\ \(F_{5}=\sum\limits_{i=1}^{n}(x_{i}^{2}-10\cos{(2\pi x_{i})}+10)\) & \(10/30/100\) & \((-5.12,5.12)^{D}\) & 0 \\ \(F_{6}=\sum\limits_{i=1}^{n}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{n}\cos{\frac{ x_{i}}{\sqrt{i}}}+1\) & \(10/30/100\) & \((-32,32)^{D}\) & 0 \\ \(F_{7}=-20\exp{\left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}} \right)}-\exp{\left(\frac{1}{n}\cos{(2\pi x_{i})}\right)}+20+e\) & \(10/30/100\) & \((-32,32)^{D}\) & 0 \\ \(F_{8}=-\frac{1+\cos(12\sqrt{x_{i}^{2}+x_{i}^{2}})}{0.5\sqrt{(x_{i}^{2}+x_{i}^{ 2})}+2}\) & \(2\) & \((-5.12,5.12)^{D}\) & \(-1\) \\ \(F_{9}=0.5+\frac{\sin^{2}(x_{i}^{2}+x_{i}^{2})-0.5}{(1+0.001(x_{i}^{2}+x_{i}^{2}) )^{2}}\) & \(2\) & \((-100,100)^{D}\) & 0 \\ \(F_{10}=(4-2.1x_{1}^{2}+\frac{x_{1}^{3}}{3})x_{1}^{2}+x_{1}x_{2}+(-4+4x_{2}^{2})x_ {2}^{2}\) & \(2\) & \((-3,3)^{D}\) & \(-1.031628\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the mathematical optimization problems. parameter is the parameter value on level \(i\). The best parameters (\(B\)-\(P\)) are listed in the last row.