## 3 Numerical Experiment
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Test Function Formula** & **Dimension** & **Search Area** & **Optimum** \\ \hline \(f_{1}\left(\widetilde{x}\right)=\Sigma_{i=1}^{n}x_{i}^{2}\). & \(3,5,10\) & \(x_{i}\in[-10,\ 10]\) & \(0\) \\ \(f_{2}\left(\widetilde{x}\right)=0.5\Sigma_{i=1}^{n}\left(x_{i}^{2}-16x_{i}^{ 2}+5x_{i}\right)\) & \(3,5,10\) & \(x_{i}\in[-5,\ 5]\) & \(-39.16599n\) \\ \(f_{3}\left(\widetilde{x}\right)=\Sigma_{i=1}^{n}\left[100\left(x_{i}+1-x_{i}^ {2}\right)^{2}+\left(x_{i}-1\right)^{2}\right)\) & \(3,5,10\) & \(x_{i}\in[-10,\ 10]\) & \(0\) \\ \(f_{4}\left(\widetilde{x}\right)=An+\Sigma_{i=1}^{n}\left(x_{i}^{2}-A\cos(2 \pi x_{i})\right)\) & \(A=10\). & \(3,5,10\) & \(x_{i}\in[-5.12,\ 5.12]\) & \(0\) \\ \(f_{5}(x,\ y)=-20\exp\left(-0.2\left(0.5\left(x^{2}+y^{2}\right)\right)^{0.5} \right)-\) & \(2\) & \(x,\ y\in[-5,\ 5]\) & \(0\) \\ \(\exp\left(0.5\left(cos2\pi x+cos2\pi y\right)\right)+\varepsilon+20\). & \(2\) & \(x,\ y\in[-10,\ 10]\) & \(0\) \\ \(f_{6}(x,\ y)=0.26\left(x^{2}+y^{2}\right)-0.48\pi y\). & \(2\) & \(x,\ y\in[-10,\ 10]\) & \(0\) \\ \(f_{7}(x,\ y)=-(y+47)\sin\left(10.5x+\left(y+47\right)\right)^{0.5}\) & \(2\) & \(x,\ y\in[-512,\ 512]\) & \(-959.6407\) \\ \(x\sin\left(\left(x-(y+47\right)\right)^{0.5}\right)\) & \(2\) & \(x,\ y\in[-10,\ 10]\) & \(0\) \\ \(f_{8}(x,y)=(x+2y-7)^{2}+(2x+y-5)^{2}\). & \(2\) & \(x,\ y\in[-10,\ 10]\) & \(0\) \\ \(f_{9}\left(\widetilde{x}\right)=-\Sigma_{i=1}^{n}\sin\left(x_{i}\right)\sin ^{20n}\left(x_{i}^{2}n^{-1}\right)\), \(m=10\). & \(2,5,10\) & \(x_{i}\in[-\pi,\ \pi]\) & \(-4.6876\) \\  & & & \(-9.6602\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test functions used to benchmark the hybrid optimization algorithms. Figure 1: (a) Dual Eggholder function \(f_{7}\) search area, which has many local extreme points. (b) A fish school optimizing the dual two-dimensional Styblinsky–Tang function \(f_{2}\). were deliberately omitted for brevity. The omitted results are similar to those shown in Table 2 and in Table 3.  
From the obtained results, shown in Table 2 and in Table 3, we see that the hybrid algorithms generally perform better than the original algorithm. To demonstrate the convergence process of the hybrid algorithms, we obtained the plots shown in Figure 2.  
As shown in Figure 2, the classical algorithms can either slightly improve the obtained solution, as shown in (a) and (d) plots, or considerably, as shown in (b) and (e) plots. The plots (c) and (f) show that the FSSN hybrid can diverge on multimodal optimization problems that can be solved fine by FSSGD. The FSSN hybrid algorithm based on [PERSON]'s method diverges in a multimodal search space in the direction towards saddle points, local minima and local maxima. The hybrid based on the collective behavior of fish schools and gradient ascent successfully converged to the optimal solution. The hybrid algorithm based on fish school search and [PERSON]'s method in optimization can show accurate results, but has limited applications. For example, the [PERSON]'s algorithm is not applicable to a function which Hessian matrix is singular, because a singular matrix is not invertible. Another limitation is the possible convergence to a wrong solution when optimizing a multimodal function. The method can converge, for example, to a saddle point. The use of the hybrid algorithm based on the collective behavior of fish schools and gradient ascent does not impose such limitations. This hybrid