# A Gradient-Based Particle-Bat Algorithm for Stochastic Configuration Network
## 4 Numerical Experiments  
### Evaluation of the G-BAPSO Algorithm  
In order to illustrate the efficiency of G-BAPSO, four benchmark functions (29)-(32) are selected to evaluate its performance, and the minimum value of each function is zero. The G-BAPSO algorithm is used to optimize these functions, and it will be compared  
Popsize, \(c_{1}\), \(c_{2}\), \(c_{3}\) and the number of iterations have an important influence on the G-BAPSO algorithm. In order to determine the optimal value, [PERSON]. [33; 34] suggested that \(c_{1}\) and \(c_{2}\) should be 2. [PERSON]. [35; 36] conducted sensitivity analysis on popsize, \(c_{1}\), \(c_{2}\) and other parameters of acceleration. Based on the benchmark test function f1, the sensitivity analysis of relevant parameters of GBAPSO is shown in Tables 1-3, and the parameters used in other experiments also follow this method. Table 1 shows the comparison results of the minimum values of the function f1 when \(c_{1}\), \(c_{2}\) and popsize are changing. It can be seen that when \(c_{1}\), \(c_{2}\) and \(c_{3}\) are fixed, the larger the popsize, the smaller the error. However, the order of magnitude of the error only changes slightly as the population size increases to 75 or 100, and the larger the popsize, the longer the calculation time consumed by the algorithm. Therefore, it is necessary to select an appropriate value that can balance the calculation time and error. Fifty is selected as the population number in solving the minimum value of \(f_{1}\). When the popsize and \(c_{3}\) are fixed, the smaller the \(c_{1}\) and \(c_{2}\) are, the smaller the error will be. When \(c_{1}\) and \(c_{2}\) drop to 0.3, the magnitude of the error has reached \(10^{-23}\), which is within the acceptable range. Table 2 shows the sensitivity analysis for \(c_{3}\). As \(c_{3}\) increases from 0.2 to 0.8, the error tends to increase first and then decrease. This is because the BAPSO algorithm plays a major role in the algorithm when \(c_{3}\) is very small, and the gradient descent algorithm plays a major role when \(c_{3}\) becomes large. When the value of \(c_{3}\) is near 0.4, there is a balance between the two, which also verifies the effectiveness of the hybrid algorithm. Table 3 shows the sensitivity analysis for the iterations. As the iterations increases, the error of the optimal solution becomes smaller and smaller. However, the more iterations, the greater the time cost. It can be seen that when iterations are 20, the error has reached the magnitude of \(10^{-23}\), which is within the acceptable range. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Popsize**} & \multicolumn{3}{c}{**The Error of the Minimum Value of ‘f1’**} \\ \cline{2-4}  & \(\mathbf{c_{1},c_{2},c_{3}=0.3,0.4}\) & \(\mathbf{c_{1},c_{2},c_{3}=1,1,0.4}\) & \(\mathbf{c_{1},c_{2},c_{3}=2,2,0.4}\) \\ \hline
25 & \(3.4575\times 10^{-23}\) & \(1.0590\times 10^{-20}\) & \(6.4554\times 10^{-7}\) \\
50 & \(1.1816\times 10^{-23}\) & \(3.1654\times 10^{-21}\) & \(1.8236\times 10^{-10}\) \\
75 & \(6.2293\times 10^{-24}\) & \(1.9501\times 10^{-21}\) & \(4.0223\times 10^{-12}\) \\
100 & \(4.5259\times 10^{-24}\) & \(1.2117\times 10^{-21}\) & \(3.8343\times 10^{-13}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The sensitivity analysis of \(c_{1}\), \(c_{2}\) and popsize. Figure 6: The figure of function \(f_{4}\). Table 4 shows the parameter Settings. The program was written on MATLAB R2019a and run on 8G memory and 1.6 Ghz CPU. Each algorithm was independently run 200 times. The experimental results are shown in Table 5.  
It can be seen from the results of four groups of experiments that G-BAPSO performs well. Compared with PSO, bat, and FOA, the optimal solution has a significant advantage in order of magnitude.