## 4 Method Validation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **Algorithm** & **Best** & **Average** & **STD** & **Variance** \\ \hline \multirow{4}{*}{\(F_{1}\)} & PSO & 0.00033 & 0.002926391 & 0.00251 & 6.302 \(\times\) 10\({}^{-6}\) \\  & MVO & 0.0060158 & 0.013780395 & 0.006517521 & 4.24781 \(\times\) 10\({}^{-5}\) \\  & WOA & \(1.88\times 10^{-85}\) & \(1.47\times 10^{-74}\) & \(2.98625\times 10^{-74}\) & \(8.9177\times 10^{-148}\) \\  & SSA & 0 & \(5.54\times 10^{-86}\) & 1.9214\(\times 10^{-85}\) & \(3.6918\times 10^{-170}\) \\  & HLSSA & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{\(F_{2}\)} & PSO & 0 & 0 & 0 & 0 \\  & MVO & 0.016637 & 0.03681265 & 0.011406764 & 0.000130114 \\  & WOA & \(9.31\times 10^{-57}\) & \(4.58\times 10^{-51}\) & \(1.31735\times 10^{-50}\) & \(1.7354\times 10^{-100}\) \\  & SSA & 0 & \(1.26\times 10^{-45}\) & \(3.79338\times 10^{-45}\) & \(1.43898\times 10^{-89}\) \\  & HLSSA & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{\(F_{3}\)} & PSO & 0.01202 & 0.12741265 & 0.1465195 & 0.021467964 \\  & MVO & 0.010296 & 0.12150025 & 0.089212352 & 0.007958844 \\  & WOA & 1.51 & \(4.33\times 10^{4}\) & 12,839.5126 & 164,853,083.9 \\  & SSA & 0 & \(1.89\times 10^{-62}\) & \(8.23118\times 10^{-62}\) & \(6.7752\times 10^{-123}\) \\  & HLSSA & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{\(F_{4}\)} & PSO & 0.01247 & 0.18308305 & 0.124116834 & 0.015404989 \\  & MVO & 0.042594 & 0.09646565 & 0.035631193 & 0.001269582 \\  & WOA & 0.132006517 & 36.21277267 & 30.75549409 & 945.9004169 \\  & SSA & 0 & \(2.10\times 10^{-48}\) & \(8.89079\times 10^{-48}\) & \(7.90461\times 10^{-95}\) \\  & HLSSA & 0 & 0 & 0 & 0 \\ \hline \multirow{4}{*}{\(F_{5}\)} & PSO & 6.3088 & 33.794385 & 41.36261671 & 1710.866061 \\  & MVO & 6.8941 & 137.409115 & 203.9184648 & 41,582.74029 \\  & WOA & 27.05372992 & 27.97991864 & 0.501056388 & 0.251057504 \\  & SSA & \(9.40\times 10^{-8}\) & \(6.21\times 10^{-5}\) & \(9.98646\times 10^{-5}\) & \(9.97294\times 10^{-9}\) \\  & HLSSA & \(2.75\times 10^{-9}\) & \(1.75\times 10^{-6}\) & \(2.57505\times 10^{-6}\) & \(6.63087\times 10^{-12}\) \\ \hline \multirow{4}{*}{\(F_{6}\)} & PSO & 0.00020952 & 0.00419352 & 0.004367111 & 1.90717 \(\times\) 10\({}^{-5}\) \\  & MVO & 0.0037676 & 0.01302205 & 0.005310819 & 2.82048 \(\times\) 10\({}^{-5}\) \\  & WOA & 0.141565425 & 0.423901093 & 0.233104106 & 0.054337524 \\  & SSA & \(6.86\times 10^{-10}\) & \(2.96\times 10^{-7}\) & \(3.99442\times 10^{-7}\) & \(1.59554\times 10^{-13}\) \\  & HLSSA & \(2.67\times 10^{-13}\) & \(3.25\times 10^{-9}\) & \(2.99447\times 10^{-9}\) & \(8.96684\times 10^{-18}\) \\ \hline \multirow{4}{*}{\(F_{7}\)} & PSO & 0.00015349 & 0.000624792 & 0.000462511 & 2.13917 \(\times\) 10\({}^{-7}\) \\  & MVO & 0.0013722 & 0.003321475 & 0.001558795 & 2.42984 \(\times\) 10\({}^{-6}\) \\ \cline{1-1}  & WOA & \(3.21249\times 10^{-5}\) & 0.005392808 & 0.005106315 & 2.60745 \(\times\) 10\({}^{-5}\) \\ \cline{1-1}  & SSA & \(0.000066649\) & 0.000400385 & 0.000294376 & 8.66569 \(\times\) 10\({}^{-8}\) \\ \cline{1-1}  & HLSSA & \(2.09\times 10^{-6}\) & \(4.91\times 10^{-5}\) & \(2.81604\times 10^{-5}\) & \(7.93008\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of optimizing seven unimodal benchmark functions using the five algorithms. In summary, the proposed HLSSA exhibited stronger optimization capabilities, faster convergence speed, and greater stability when dealing with unimodal functions. This study proposes an efficient and reliable algorithm for unimodal test function optimization problems. Figure 4: (**a**) Three-dimensional plot of \(F_{3}\); (**b**) convergence curve. Figure 5: (**a**) Three-dimensional plot of \(F_{4}\); (**b**) convergence curve. Figure 3: (**a**) Three-dimensional plot of \(F_{2}\); (**b**) convergence curve.