## 5 CGE-SSA Multi-Objective Optimization
### Improved Sparrow Algorithm
#### 5.2.4 Simulation Verification  
In order to verify the effectiveness of CGE-SSA, six test functions (three single-peak functions and three multi-peak functions) were selected to test CGE-SSA's global solving ability and the ability to move away from the local optimum by comparing it with the SSA, the genetic algorithm (GA), and the particle swarm algorithm (PSO). The test functions are described in Table 6.  
The results of 50 experiments are shown in Table 7 and Figure 11. In Figure 11a-c, all three functions are single-peaked functions, so the iteration curves of the algorithms are closer. In Figure 11a-d, it can be seen that the iterative curves produced using the four functions of CGE-SSA converge at the 22 nd, 21 st, 25 th, and 28 th generations, respectively. The four function iteration curves produced using SSA do not converge, and the last generation results are \(2.3\times 10^{-67}\), \(1.6\times 10^{-30}\), \(3.09\times 10^{-41}\), and \(-7874.28\). The \(f_{1}\),\(f_{2}\), and \(f_{3}\) function curves produced using GA converge, and the convergence results are 30,717, 69.7, and 59,675. The \(f_{1}\) and \(f_{2}\) function curves produced using PSO also converge, and the convergence results are 21.3 and 20.06. As can be seen in Figure 11c,d, the convergence speed and accuracy of the CGE-SSA iterative curves are significantly higher than those of other algorithms.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Function** & **Dim** & **Range** \\ \hline \(f_{1}=\sum\limits_{i=1}^{n}x_{i}^{2}\) & 30 & \([-100,100]\) \\ \(f_{2}=\sum\limits_{i=1}^{n}|x_{i}|+\prod\limits_{i=1}^{n}|x_{i}|\) & 30 & \([-10,10]\) \\ \(f_{3}=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{i}x_{j}\right)^{2}\) & 30 & \([-100,100]\) \\ \(f_{4}=\sum\limits_{i=1}^{n}-x_{i}\text{sin}\big{(}\sqrt{|x_{i}|}\big{)}\) & 30 & \([-500,500]\) \\ \(f_{5}=(\frac{1}{500}+\frac{25}{j=1}\frac{1}{j+\sum\limits_{i=1}^{2}\big{(}x_{ i}-a_{ij}\big{)}^{6}})^{-1}\) & 2 & \([-65.536,65.536]\) \\ \(f_{6}=-\sum\limits_{i=1}^{5}\big{[}(x-a_{i})(x-a_{i})^{T}+c_{i}\big{]}^{-1}\) & 4 & \([0,4]\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Test functions.  
Figure 10: Dynamic weight change curves.  
The comparison of the convergence curves of each test function and the experimental results show that CGE-SSA has a better convergence speed and accuracy than SSA, GA, and PSO, and the standard deviation and mean of optimal fitness are smaller; the optimization effect of CGE-SSA is far better than those of the other algorithms, especially in the single-peak function, thus reflecting the superiority and stability of CGE-SSA. In summary, CGE-SSA shows improvement in terms of both optimization search time and global search ability compared with other algorithms.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **Parameter** & **SSA** & **CGE-SSA** & **GA** & **PSO** \\ \hline \multirow{2}{*}{\(f_{1}\)} & AVG & \(2.3\times 10^{-67}\) & 0 & 30,717 & 21.3 \\  & STD & \(1.2\times 10^{-66}\) & 0 & 8176.9 & 2.63 \\  & AVG & \(1.6\times 10^{-30}\) & 0 & 69.7 & 20.06 \\  & STD & \(1.1\times 10^{-29}\) & 0 & 10.7 & 1.9 \\  & AVG & \(3.09\times 10^{-41}\) & 0 & 59,675 & 621.9 \\  & STD & \(2.12\times 10^{-40}\) & 0 & 19,941 & 296.2 \\  & AVG & \(-7874.28\) & \(-8500.69\) & \(-2286\) & \(-3989\) \\  & STD & \(604.83\) & \(517.46\) & \(464.9\) & \(417.6\) \\  & AVG & \(4.66\) & \(1.4\) & \(1.63\) & \(2.34\) \\  & STD & \(5.1\) & \(0.83\) & \(1.38\) & \(1.93\) \\  & AVG & \(-7.90\) & \(-10.1532\) & \(-0.8\) & \(-4.14\) \\  & STD & \(2.55\) & \(7.76\times 10^{-7}\) & 0.43 & 2.4 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of experimental results.