## 4 Experimental Analysis
### Simulation Results on CEC2013 Standard Bounded Constraint Benchmark  
In the following experimental results, we used the CEC2013 benchmark function set to verify the performance of our newly proposed MQSOS algorithm. There are 5 single objective functions (\(f_{1}\)-\(f_{5}\)), 15 multi-objective functions (\(f_{6}\)-\(f_{20}\)), and 8 composite functions (\(f_{21}\)-\(f_{28}\)) in the CEC2018 benchmark function set. The performance of the MQSOS algorithm was verified by comparing the minimum values obtained by the MQSOS algorithm with the PSO, PPSO (uses the algorithm's first parallel strategy), APSO, SOS, OSOS, and QUATRE/best/1 algorithms in the 28 functions of CEC2013. These algorithms were set to a 10-dimensional search space with a search range of \(-100\) to 100 for each dimension and 100 individual organisms in the population. The averages of 51 runs obtained by running each of the algorithms in the 28 test functions of CEC2013 are shown in Table 2.  
In Table 2, the bolded values represent the smallest fitness function error values obtained by comparing the seven algorithms. The MQSOS algorithm obtained a better fitness function value than other algorithms when using 12 benchmark functions (\(f_{3}\), \(f_{6}\), \(f_{12}\), \(f_{13}\), \(f_{14}\), \(f_{15}\), \(f_{18}\), \(f_{22}\), \(f_{23}\), \(f_{26}\), \(f_{27}\), \(f_{28}\)). OSOS achieved the optimal value using four benchmark functions (\(f_{7}\), \(f_{11}\), \(f_{24}\), \(f_{25}\)), QUATRE/best/1 in three benchmark functions (\(f_{1}\), \(f_{5}\), \(f_{10}\)), the SOS algorithm in two benchmark functions (\(f_{9}\), \(f_{20}\)), and APSO in seven benchmark functions (\(f_{2}\), \(f_{4}\), \(f_{8}\), \(f_{16}\), \(f_{17}\), \(f_{19}\), \(f_{20}\)). PSO and PPSO did not get the optimal value in any of the 28 benchmark functions.  
The global optimization capabilities of the seven algorithms based on the 28 benchmark functions of CEC2013 are shown in Figures 2-5. According to the global optimization process and the final  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**10D** & **QSOS** & **OSOS** & **QUATRE/best/1** & **SOS** & **PSO** & **PSO** & **APSO** \\ \hline \(f_{1}\) & \(1.18\times 10^{-10}\) & 0.009640731 & \(\mathbf{2.27\times 10^{-13}}\) & 0.038254473 & 289.4136577 & 10.04216027 & \(5.62\times 10^{-13}\) \\ \(f_  
optimization result of each algorithm in these figures, the MQSOS algorithm was superior to the other algorithms in 15 benchmark functions (\(f_{8}\), \(f_{9}\), \(f_{11}\), \(f_{12}\), \(f_{13}\), \(f_{14}\), \(f_{15}\), \(f_{16}\),\(f_{18}\), \(f_{20}\), \(f_{22}\), \(f_{23}\), \(f_{24}\), \(f_{26}\), \(f_{28}\)). The APSO algorithm outperformed other algorithms in 8 benchmark functions (\(f_{1}\), \(f_{2}\), \(f_{3}\), \(f_{5}\), \(f_{10}\), \(f_{17}\), \(f_{19}\), \(f_{21}\)). The QUATRE/best/1 algorithm outperformed other algorithms in 3 benchmark functions (\(f_{4}\), \(f_{7}\), \(f_{27}\)). The SOS algorithm obtained the optimal value in benchmark function \(f_{6}\). The OSOS algorithm achieved this in benchmark function \(f_{25}\).  
Figure 2: Comparison of the best fitnesses for functions \(f_{1}-f_{8}\) with 10D optimization.  
Figure 3: Comparison of the best fitnesses for functions \(f_{9}-f_{16}\) with 10D optimization.  
Figure 4: Comparison of the best fitnesses for functions \(f_{17}-f_{28}\) with 10D optimization.