## 3 Methodology
### Iajs-Hkrvm Model
#### 3.1.2 Improved Artificial Jellyfish Search Algorithm
Table 2 shows the statistical results of the 30 independent tests of each algorithm on the benchmark test functions, where the bold part represents the algorithm with the best optimization performance. Figure 4 shows the average iterative convergence curves on the benchmark test functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Algorithm Name**} & \multicolumn{1}{c}{**Parameters**} \\ \hline IAJS & \(\beta=3\), \(\gamma=0.1\), \(k_{\max}=10\), \(k_{\min}=1\) \\ AJS & \(\beta=3\), \(\gamma=0.1\) \\ GWO & \(a_{\max}=2\), \(a_{\min}=0\), \(r_{1}\), \(r_{2}\in[0,1]\) \\ WOA & \(a\in[0,2]\), \(r_{1}\), \(r_{2}\in[0,1]\), \(p=0.5\), \(b=1\), \(l\in[-1,1]\) \\ HHO & \(p=0.5\), \(l\in[0,2]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The basic parameter settings of each algorithm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Function**} & \multicolumn{1}{c}{**Statistics**} & \multicolumn{1}{c}{**Algorithm**} & \multicolumn{1}{c}{**Algorithm**} \\ \cline{3-8} \multicolumn{2}{c}{} & **IAJS** & **AJS** & **GWO** & **WOA** & **HHO** \\ \hline \multirow{3}{*}{F1: Sphere} & Optimal & \(\mathbf{0.0000}\times 10^{-00}\) & \(2.9132\times 10^{-19}\) & \(4.4427\times 10^{-29}\) & \(3.0818\times 10^{-86}\) & \(3.4871\times 10^{-111}\) \\  & Mean & \(\mathbf{0.0000}\times 10^{-00}\) & \(1.5817\times 10^{-17}\) & \(1.9413\times 10^{-27}\) & \(2.2725\times 10^{-74}\) & \(6.6443\times 10^{-96}\) \\  & Standard & \(\mathbf{0.0000}\times 10^{-00}\) & \(1.8679\times 10^{-17}\) & \(2.8171\times 10^{-27}\) & \(1.0694\times 10^{-73}\) & \(3.5842\times 10^{-95}\) \\ \hline \multirow{3}{*}{F2: Schewefel 1.2} & Optimal & \(\mathbf{0.0000}\times 10^{-00}\) & \(1.9518\times 10^{-18}\) & \(7.6909\times 10^{-09}\) & \(2.1240\times 10^{-01}\) & \(2.4287\times 10^{-97}\) \\  & Mean & \(\mathbf{0.0000}\times 10^{-00}\) & \(7.2734\times 10^{-17}\) & \(1.1741\times 10^{-50}\) & \(8.1110\times 10^{-10}\) & \(1.2096\times 10^{-79}\) \\  & Standard & \(\mathbf{0.0000}\times 10^{-00}\) & \(8.7846\times 10^{-17}\) & \(2.4667\times 10^{-5}\) & \(3.6620\times 10^{-3}\) & \(3.6922\times 10^{-79}\) \\ \hline \multirow{3}{*}{F3: Ackley} & Optimal & \(\mathbf{8.8818}\times 10^{-16}\) & \(1.0691\times 10^{-10}\) & \(7.5495\times 10^{-14}\) & \(8.8818\times 10^{-16}\) & \(8.8818\times 10^{-16}\) \\  & Mean & \(\mathbf{8.8818}\times 10^{-16}\) & \(4.6871\times 10^{-10}\) & \(1.0640\times 10^{-13}\) & \(3.9672\times 10^{-15}\) & \(8.8818\times 1 As shown in Table 2, the optimization results of the IAJS are superior to the AJS, WOA, GWO, and HHO. The optimal value and average value of the IAJS test results on F1-F2 and F4 can reach the theoretical optimal value, and the standard deviation is 0, but the theoretical optimal value is not reached on F3, because there are many local optimal values in F3, making it easy to fall into the local optimal value. However, the standard deviation of the experiment is 0. Therefore, the optimization performance of the IAJS algorithm has been greatly improved. In addition, it can be seen from Figure 4 that the IAJS algorithm converges faster than the other algorithms, indicating that the IAJS algorithm can jump out of the local optimal faster. Meanwhile, it can be seen from Figure 4a,b that compared with the other four algorithms, the convergence speed and accuracy of the IAJS algorithm are significantly improved. Additionally, it can be seen from Table 2 and Figure 4d that all five algorithms have reached the theoretical optimal solution, but from the convergence curves, the convergence speed of the IAJS algorithm is faster than that of the other four algorithms. Therefore, the AJS algorithm improved by Fuch chaotic mapping, lens-imaging reverse learning, and adaptive hybrid mutation strategy can effectively improve the optimization accuracy and efficiency. Thus, it will be applied to the kernel parameter optimization of the HKRVM model.