## 3 Methodology
### Synergistic Adaptive Genetic Algorithm
#### 3.1.3 Optimization Experiments  
In order to test the optimization performance of SAGA, compared with H-SGA [36], IAGA [37], and MAGA [38], we select several classical functions to carry out the comparing experiments. These functions are shown as below:  
Figure 3: The adaptive adjustment curve of \(P_{c}\).  
Figure 2: The illustration of the adjustment process.  
\(F_{1}\): One-dimension multimodal function.  
\[\begin{split} F_{1}=x+10\sin(5x)+7\cos(4x),\\ 0\leqslant x\leqslant 9\end{split} \tag{14}\]  
This function has multiple local optima in \(x{\in}[0,9]\), and the only one maximum is \(F_{1}\) (\(7.8568\)) = 24.8554. The graph of \(F_{1}\) is shown in Figure 5.  
\(F_{2}\): Camel function.  
\[\begin{split} F_{2}=(4-2.1x_{1}^{2}+\frac{x_{1}^{2}}{3})x_{1}^{2} +x_{1}x_{2}+(-4+x_{2}^{2})x_{2}^{2},\\ -3\leqslant x_{i}\leqslant 3,i=1,2\end{split} \tag{15}\]  
This function has six local minima and two global minima, which are \(F_{2}(-0.0898,0.7126)\) = \(F_{2}(0.0898,-0.7126)\) = \(-1.031628\). The graph of \(F_{2}\) is shown in Figure 6.  
\(F_{3}\): Schaffer's \(f_{6}\)[39].  
\[\begin{split} F_{3}=0.5+\frac{\sin^{2}\sqrt{x_{1}^{2}+x_{2}^{2}} -0.5}{[1.0+0.001(x_{1}^{2}+x_{2}^{2})]^{2}},\\ -100\leqslant x_{i}\leqslant 100,i=1,2\end{split} \tag{16}\]  
The function is symmetric about the origin and has multiple local optimal values. Its global minimum is \(F_{3}(0,0)\) = 0. As a result, this function is very suitable for optimization experiment. The graph of \(F_{3}\) is shown in Figure 7.  
Figure 4: The adaptive adjustment curve of \(P_{m}\).  
Figure 5: The graph of function \(F_{1}\).  
\(F_{4}\): Schaffer's \(f_{7}\)[39].  
\[F_{4}=(x_{1}^{2}+x_{2}^{2})^{0.25}[\sin^{2}(50(x_{1}^{2}+x_{2}^{2})^{0.1})+0.1],\] \[-100\leqslant x_{i}\leqslant 100,i=1,2 \tag{17}\]  
This function is similar to \(F_{3}\), and its global minimum is \(F_{4}(0,0)=0\). The graph of \(F_{4}\) is shown in Figure 8.  
Figure 8: The graph of function \(F_{4}\).  
Figure 6: The graph of function \(F_{2}\).  
Figure 7: The graph of function \(F_{3}\).  
In all experiments, the size of population is 20, and the binary length is 22. The number of iterations in \(F_{1}\) is 200, and that of in other function is 600. In H-SGA [36], \(P_{c}\) is altering from 0.5 to 0.75, and \(P_{m}\) is altering from 0.05 to 0.08. The range of \(P_{c}\) in IAGA [37] and SAGA is (0.6, 0.8), and that of \(P_{m}\) is (0.05, 0.08). In MAGA [38], \(P_{c1}\) = 0.7, \(P_{c2}\) = 0.2, \(P_{m1}\) = 0.07, \(P_{m2}\) = 0.03, \(\gamma\) = 2 and \(\alpha\) = 0.08.  
Based on the above conditions, each function is executed 1000 times independently with four algorithms in an initial environment randomly. In addition, the calculation accuracy of function \(F_{1}\) and \(F_{2}\) is \(10^{-4}\), and that of \(F_{3}\) and \(F_{4}\) is \(10^{-6}\). Tables 1-3 list the experimental results, for example, the number of convergences, average convergence generation, and value.  
From Tables 1 and 2, we can easily find that SAGA and MAGA [38] have higher convergence rates and fewer iterations compared with H-SGA [36] and IAGA [37], which indicates these two methods have obvious advantages in convergence speed. However, as shown in Table 3, SAGA is better than MAGA [38] in the final convergence result, which fully proves the method proposed in this paper has outstanding global optimization ability. Figures 9-12 are the experimental results of the comparison algorithm on each optimization function.  
Figure 11: The experimental results on function \(F_{3}\).  
Figure 12: The experimental results on function \(F_{4}\).  
Figure 9: The experimental results on function \(F_{1}\).