## 3 Experiment and Application
### Performance Analysis: Classic Benchmark Functions
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Test Function** & **D** & **Range** & **Optimum** \\ \hline \(f_{6}(x)=\sum_{i=1}^{n}\left[x_{i}+0.5\right]\right)^{2}\) & 30 & \(\left[-100,\text{100}\right]\) & 0 \\ \hline \(f_{7}(x)=\sum_{i=1}^{n}ix_{i}^{4}+random[0,1)\) & 30 & \(\left[-1.28,\text{1.28}\right]\) & 0 \\ \hline \(f_{8}(x)=\sum_{i=1}^{n}-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & 30 & \(\left[-500,\text{500}\right]\) & \(-418.9825\text{*}5\) \\ \hline \(f_{9}(x)=\sum_{i=1}^{n}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & 30 & \(\left[-5.12,5.12\right]\) & 0 \\ \hline \(f_{10}(x)=-20 exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}\right)\) & 30 & \(\left[-32,\text{32}\right]\) & 0 \\ \(-exp\left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)+20+e\) & & & \\ \hline \(f_{11}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos\left( \frac{x_{i}}{\sqrt{i}}\right)+1\) & 30 & \(\left[-600,\text{600}\right]\) & 0 \\ \hline \(f_{12}(x)=\frac{n}{4}[10\sin(\pi y_{1})\) & & & \\ \(+\sum_{i=1}^{n-1}\left(y_{i}-1\right)^{2}\left[1+10 sin^{2}(\pi y_{i+1})\right]\) & & \\ \(+(y_{n}-1)^{2}\right]+\sum_{i=1}^{n}u(x_{i},10,100,4)\) & & \\ \(y_{i}=1+\frac{x_{i}+1}{4}\) & 30 & \(\left[-50,\text{50}\right]\) & 0 \\ \(u(x_{i},a,k,m)=\left\{\begin{array}{c}k(x_{i}-a)^{m}\ x_{i}>a\\ 0-a<x_{i}<a\\ k(-x_{i}-a)^{m}\ x_{i}<-a\end{array}\right.\) & & \\ \hline \(f_{13}(x)=0.1\left[sin^{2}(3\pi x_{i})\right.\) & & \\ \(+\sum_{i=1}^{n}(x_{i}-1)^{2}\left[1+sin^{2}(3\pi x_{i}+1)\right]\) & 30 & \(\left[-50,\text{50}\right]\) & 0 \\ \(+(x_{n}-1)^{2}\left[1+sin^{2}(2\pi x_{n})\right]\right]\) & & \\ \(+\sum_{i=1}^{n}u(x_{i},5,100,4)\) & & & \\ \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \(f_{14}(x)=\left(\frac{1}{500}+\sum_{j=1}^{25}\frac{1}{j+\sum_{j=1}^{2}(x_{i}- a_{ij})^{4}}\right)^{-1}\) & 2 & \(\left[-65,\text{65}\right]\) & 1 \\ \hline \(f_{15}(x)=\sum_{i=1}^{11}\left[a_{i}-\frac{x_{1}\left(y_{i}^{2}+b_{i}y_{2} \right)}{b_{i}^{2}+bIn this test, the population size \(N\) was set to 30. The maximum function evolution numbers (\(T\)) was set to 500. For each test function, 30 independent runs were carried out with random seeds. The parameter settings for the other compared algorithms in this test are detailed in Table 4. Such values represent the best parameter sets for these algorithms according to the original paper of [PERSON] [8], WOA [9], PSO [2], and DE [68]. Table 5 summarizes the statistical results including mean, worst, best and SD of objective function values obtained by 30 independent runs for high-dimensional functions in Table 2. Compared with the other four algorithms, APO exhibited the best performance on the high-dimensional benchmark functions except function \(f_{4}\). Gray wolf optimization had a promising performance for optimizing \(f_{4}\), and APO exhibited the second-best performance on this function; meanwhile, APO sometimes has the best value for optimizing \(f_{4}\). From Table 5, it can be found that APO nearly finds all the best values of \(f_{9}\) and \(f_{11}\). According to the test results of Table 5, APO outperforms the other algorithms on high-dimensional benchmark functions. \begin{table}
\begin{tabular}{l l} \hline \hline
**Algorithms** & **Parameter Settings** \\ \hline Gray wolf optimization (GWO) [8] & \(a=2-t^{*}2/T\) \\ \hline Whale optimization algorithm (WOA) [9] & \(a_{1}=2-t^{*}2/T\), \(a=-1+t^{*}(-1)/T\) \\ \hline Particle swarm optimization (PSO) [2] & \(w_{max}=0.9\), \(w_{min}=0.2\), \(c_{1}=c_{2}=2\) \\ \hline Differential evolution (DE) [68] & \(F=0.5\), \(CR=0.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameter settings for the other compared algorithms.