## 3 Numerical Experiments and Analysis  
### Benchmark Function  
In the process of developing and improving optimization algorithms, benchmark function testing is highly valued [7]. Twelve benchmark functions are selected for the  
Figure 1: Flowchart of the WaOA.  
simulation test, as shown in Table 1. The definition domain limits the search space, and the theoretical values reflect the optimal solutions. F1-F6 are unimodal functions that possess only one global optimal solution, without a local optimum. They are utilized to test the convergence speed and development behavior of the algorithm. F7-F12 are multimodal functions, which possess global optimum as well as several local optimal solutions. The number grows exponentially as the dimension increases, which can be utilized to examine the capability of the algorithm to break away from local optimal and global search capability [8; 9]. One of the advantages of the multimodal function is that it has high similarity with the search space of practical problems [4]. Figure 2 displays the diagram of the search space of the WaOA in benchmark functions F7-F12.  
The theoretical optimal value of function F8 presents the feature tightly related to its dimension. Meanwhile, it changes along with the change of the dimension. The ideal value of F8 ought to be \(f_{\text{min}}=-418.9872\times\text{dim}\). When the dimension denotes 30, the optimal value should be \(-12  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Number** & **Name** & **Benchmark Functions** & **Dim** & **Range** & \(f_{\text{min}}\) \\ \hline
**F1** & Sphere & \(f_{1}(x)=\sum\limits_{i=1}^{Dim}x_{i}^{2}\) & 30 & [\(-100\),100] & 0 \\
**F2** & Schwefel 2.22 & \(f_{2}(x)=\sum\limits_{i=1}^{Dim}|x_{i}|+\prod\limits_{i=1}^{Dim}|x_{i}|\) & 30 & [\(-10\),10] & 0 \\
**F3** & Schwefel 1.2 & \(f_{3}(x)=\sum\limits_{i=1}^{Dim}\sum\limits_{i=1}^{2}(\sum\limits_{i=1}^{2}x_{ i})\) & 30 & [\(-100\),100] & 0 \\
**F4** & Schwefel 2.21 & \(f_{4}(x)=\max_{i}\{|x_{i}|,1\leq i\leq D\}\) & 30 & [\(-100\),100] & 0 \\
**F5** & Rosenbrock & \(f_{5}(x)=\sum\limits_{i=1}^{Dim-1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^ {2}\right]\) & 30 & [\(-30\),30] & 0 \\
**F6** & Step & \(f_{6}(x)=\sum\limits_{i=1}^{Dim}\left[(x_{i}+0.5)\right]\) & 30 & [\(-100\),100] & 0 \\
**F7** & Quartic & \(f_{7}(x)=\sum\limits_{i=1}^{Dim}i\times x_{i}^{4}+random[0,1)\) & 30 & [\(-1.28\),1.28] & 0 \\
**F8** & Schwefel 12.26 & \(f_{8}(x)=\sum\limits_{i=1}^{Dim}-x_{i}\sin\bigl{(}\sqrt{x_{i}}\bigr{)}\) & 30 & [\(-500\),500] & \(-12\),569.5 \\
**F9** & [PERSON] & \(f_{9}(x)=\sum\limits_{i=1}^{Dim}\left[x_{i}^{2}-10\cos(2\pi x_{i}+10)\right]\) & 30 & [\(-5.12\),5.12] & 0 \\
**F10** & Ackley & \(f_{10}(x)=-20\exp\left(-0.2\sqrt{\frac{i}{Dim}\sum\limits_{i=1}^{Dim}x_{i}^{2}}\right)\) & 30 & [\(-32\),32] & 0 \\  & & \(-\exp\left(\frac{1}{Dim}\sum\limits_{i=1}^{Dim}\cos(2\pi x_{i})\right)+20+e\) & & & \\
**F11** & [PERSON] & \(f_{11}(x)=\frac{1}{4000}\sum\limits_{i=1}^{Dim}x_{i}^{2}-\prod\limits_{i=1}^{ Dim}\cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & 30 & [\(-600\),600] & 0 \\
**F12** & Generalized penalized & \(f_{12}(x)=\frac{\pi}{Dim}\left\{10\sin(\pi y_{1})+\sum\limits_{i=1}^{Dim}(y_{ i}-1)^{2}[1+10\sin^{2}(\pi y_{i+1})]\right.\) & & \\  & & \(+(y_{Dim}-1)^{2}\right\}+\sum\limits_{i=1}^{Dim}n(x_{i},10,100,4)\) & 30 & [\(-50\),50] & 0 \\  & & \(y_{i}=1+\frac{x_{i}+1}{4}\), & & & \\  & & \(u(x_{i},a,k,m)=\begin{cases}k(x_{i}-a)^{m}&x_{i}>a\\ 0&-a<x_{i}<a\\ k(-x_{i}-a)^{m}&x_{i}<-a\end{cases}\) & & & \\ \hline \end{tabular}
\end{table}
Table 1: Test function.  
Regarding the function F12,  
\[\begin{split}& y_{i}=1+\frac{x_{i}+1}{4},\\ & u(x_{i},a,k,m)=\begin{cases}k(x_{i}-a)^{m}&x_{i}>a\\ 0&-a<x_{i}<a\\ k(-x_{i}-a)^{m}&x_{i}<-a\end{cases}\end{split} \tag{10}\]