## 4 Benchmark Function Testing and Analysis
### Comparison of ADGMPSO to Other Mainstream Evolutionary Algorithms  
For the comparison experiments with the proposed ADGMPSO, we selected five existing algorithms: the particle swarm optimization algorithm (PSO), butterfly optimization algorithm (BOA) [30], whale optimization algorithm (WOA) [31], squirrel search algorithm (SSA) [32], and gray wolf optimization algorithm (GWO) [33]. We analyzed the performance of the algorithms based on the obtained optimal values, mean values, and standard deviations. The population size of each algorithm was set to 40, and the maximum number of iterations was set to 2000. All the algorithms were randomly initialized, with the exception of ADGMPSO, which utilized the cat chaos initialization population. The termination condition for all the algorithms was that the current iteration count reached the maximum number of iterations. The essential parameters for each algorithm were set, as shown in Table 4.  
To minimize the experimental error caused by the randomness, 20 comparative experiments were conducted on 10 benchmark functions for each of the six algorithms, including the proposed ADGMPSO, using the same simulation equipment and operating environment specified in Table 4. Table 5 displays the best results, average results, and standard deviations of the various methods on the ten benchmark functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameter** \\ \hline PSO & \(\omega=0.9\), \(c_{1}=c_{2}=2\) \\ BOA & C = 0.01, P = 0.8, \(\alpha\) increases linearly from 0.1 to 0.3 \\ WOA & r\({}_{1}\), r\({}_{2}\)\(\in\) [0, 1], \(\alpha\) decreases linearly from 2 to 0 \\ SSA & P\({}_{\mathrm{dp}}=0.1\), G\({}_{\mathrm{c}}\)\(=1.9\), sf = 18 \\ GWO & r\({}_{1}\), r\({}_{2}\)\(\in\) [0, 1], \(\alpha\) decreases linearly from 2 to 0 \\ ADGMPSO & \(\omega_{max}=0.9\), \(\omega_{min}=0.2\), \(c_{max}=2\), \(c_{min}=1\), \(\gamma_{min}=0.2\), \(\gamma_{max}=1\), \(\gamma\) \\  & decreases linearly from 1 to 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings for the different algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Func.** & **Name** & **Search Range** & **Optimal Value** & **Dim.** \\ \hline \(f_{1}\) & Sphere & \([-100,100]\) & 0 & 30 \\ \(f_{2}\) & Schwefel 2.22 & \([-10,10]\) & 0 & 30 \\ \(f_{3}\) & Schwefel 1.2 & \([-100,100]\) & 0 & 30 \\ \(f_{4}\) & Step & \([-100,100]\) & 0 & 30 \\ \(f_{5}\) & Schwefel 2.21 & \([-100,100]\) & 0 & 30 \\ \(f_{6}\) & Quartic & \([-1.28,1.28]\) & 0 & 30 \\ \(f_{7}\) & Rastrigin & \([-5.12,5.12]\) & 0 & 30 \\ \(f_{8}\) & Ackley & \([-32,32]\) & 0 & 30 \\ \(f_{9}\) & Griewank & \([-600,600]\) & 0 & 30 \\ \(f_{10}\) & Penalized & \([-50,50]\) & 0 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Benchmark test functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Running Environment** & **Details** \\ \hline Operating System & Windows 10 \\ CPU & I5 9300H [EMAIL_ADDRESS] GHz \\ RAM & 16 GB DDR4 RAM \\ Software & MATLAB2016a \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Benchmark function running environment table. It can be inferred from the experimental findings of the benchmark test functions reported in Table 5 that ADGMPSO identified the theoretical optimal solution in terms of the optimal values for solving the single-peak functions \(f_{1}\), \(f_{2}\), \(f_{3}\), as well as the multi-peak functions \(f_{7}\) and \(f_{9}\). Even though ADGMPSO failed to reach the theoretically ideal value for the test functions \(f_{4}\), \(f_{5}\), \(f_{6}\), and \(f_{10}\) when determining the ideal value, it significantly outperformed the other five algorithms by orders of magnitude. The optimal value alone did not reflect the overall algorithm performance. However, upon analyzing the average results presented in Table 5, it became apparent that ADGMPSO consistently achieved the lowest mean value across all ten benchmark test functions compared to the other five algorithms.