## 4 Experiments and Results
### Evaluation of Exploration Capability (Functions F8-F23)  
The multimodal functions have several local optimums determined by the problem size (i.e., decision variables), where the number of local optimums increases with increasing the size of the problem. Accordingly, the multimodal functions play the primary role in evaluating the optimization algorithm's exploration capability. The results presented in Table S2 prove the demonstration of the proposed LO against the compared algorithms, where the LO obtains the best results in achieving the best in ten functions, including F8, F10, F11, F14, F21, and F23, and Worst and Mean in 10 functions, including F10, F14, F16-F19, and F23. These results prove the LO's robust performance in managing its exploration capability, which significantly leads LO towards the global optimum. The detailed results can be found in the Supplemental Information in Tables S1 and S2.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Function** & **Test Functions** & **Range** & \(n\) & **C** & \(f(\mathbf{x^{\star}})\) \\ \hline F14 & \(\left(\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+2^{2}_{1}(\gamma-z_{ij})^{2}} \right)^{-1}\) & \(x_{i}\in[-65,65]\) & 2 & M & 1 \\ F15 & \(\sum_{i=1}^{11}\left[a_{i}-\frac{x_{i}(\gamma_{i}^{2}+b\gamma_{ij})}{B_{i}^{2 }+b\gamma_{ij}+\gamma_{ij}+4}\right]^{2}\) & \(x_{i}\in[-5,5]\) & 4 & M & 0.00030 \\ F16 & \(4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^{4}\) & \(x_{i}\in[-5,5]\) & 2 & M & \(-1.0316\) \\ F17 & \(\left(x_{2}-\frac{51}{50}x_{1}^{2}+\frac{5}{5}x_{1}-6\right)^{2}+10\left(1- \frac{1}{50t}\right)\cos x_{1}+10\) & \(x_{i}\in[-5,5]\) & 2 & M & 0.398 \\ F18 & \(\left[1+(x_{1}+x_{2}+1)^{2}(19-14x_{1}+3x_{1}^{2}-14x_{2}+6x_{1}x_{2}+3x_{2}^{ 2})\right]\) & \(x_{i}\in[-2,2]\) & 2 & M & 3 \\ F19 & \(-\sum_{i=1}^{4}c_{i}\exp\left(-\sum_{j=1}^{4}a_{ij}(x_{j}-p_{ij})^{2}\right)\) & \(x_{i}\in[1,3]\) & 3 & M & \(-3.86\) \\ F20 & \(-\sum_{i=1}^{4}c_{i}\exp\left(-\sum_{j=1}^{6}a_{ij}(x_{j}-p_{ij})^{2}\right)\) & \(x_{i}\in[0,1]\) & 6 & M & \(-3.32\) \\ F21 & \(-\sum_{i=1}^{4}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \(x_{i}\in[0,10]\) & 4 & M & \(-10.1532\) \\ F22 & \(-\sum_{i=1}^{4}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \(x_{i}\in[0,10]\) & 4 & M & \(-10.4028\) \\ F23 & \(-\sum_{i=1}^{40}\left[(X-a_{i})(X-a_{i})^{T}+c_{i}\right]^{-1}\) & \(x_{i}\in[0.10]\) & 4 & M & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The characteristics of fixed-dimension multimodal benchmark functions.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Function** & **Test Functions** & **Range** & \(n\) & **C** & \(f(\mathbf{x^{\star}})\) \\ \hline F8 & \(\sum_{i=1}^{n}-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & \(x_{i}\in[-500,500]\) & 30 & M & \(-12\),569.5 \\ F9 & \(\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & \(x_{i}\in[-5.12,5.12]\) & 30 & M & 0 \\ F10 & \(-20\exp(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}\chi_{i}^{2}})-\exp\left(\frac{1}{n} \sum_{i=1}^{n}cos(2\pi x_{i})\right)+20+\epsilon\) & \(x_{i}\in[-32,32]\) & 30 & M & 0 \\ F11 & \(\frac{1}{400}\sum_{i=1}^{n}\chi_{i}^{2}-\prod_{i=1}^{n}\cos\left(\frac{x_{i}}{ \sqrt{i}}\right)+1\) & \(x_{i}\in[-600,600]\) & 30 & M & 0 \\ F12 & \(\frac{1}{n}\left[10\sin(\pi y_{1})+\sum_{i=1}^{n}(y_{i}-1)^{2}\left[1+10\sin^{2}( \pi y_{i+1})\right]+(y_{n}-1)^{2}\right\}+\sum_{i=1}^{n}u(x_{i},10,100,4)\) & \(x_{i}\in[-50,50]\) & 30 & M & 0 \\  & \(y_{i}=1+\frac{x_{i}+1}{4}u(x_{i},a,k,m)=\begin{cases}k(x_{i}-a)^{m}&x_{i}>a\\ 0-a&<x_{i}<a\\ k(-x_{i}-a)^{m}&x_{i}<-a\end{cases}\) & \(+\) & \(+\) & \(+\) & \(+\) \\ F13 & \(0.1\!\left\{\sin^{2}(3\pi x_{i})+\sum_{i=1}^{n}(x_{i}-1)^{2}\left[1+\sin^{2}(3 \pi x_{i}+1)\right]+(x_{n}-1)^{2}\left[1+\sin^{2}(2\pi x_{n})\right]\right\}\) & \(+\) & \(+\) & \(+\) & \(+\) \\ F13 & \(\sum_{i=1}^{n}u(x_{i},5,100,4)\) & \(+\) & \(+\) & \(+\) & \(+\) & \(+\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The characteristics of multimodal benchmark functions.