## 4 Experimental Results
### Performance Comparison  
#### 4.2.1 Experiment 1: Impact of Interaction Operation  
This experiment compares ICAAI and ICACI against PSO and Perturbed ICA [35]. Wilcoxon signed-rank test has been conducted between every two methods, and \(p\)-values less than 0.05 are deemed statistically significant. Table 2 shows the average and the standard deviation of the best objective values obtained by each method after 1000 generations. Since ICAAI and ICACI are essentially the Perturbed ICA with the addition of an interaction step (line 10 of Figure 2), the impact of the interaction step can be shown by comparing the performance of ICAAI and ICACI against that of the Perturbed ICA. Wilcoxon signed-rank test shows that ICAAI statistically significantly outperforms the Perturbed ICA on benchmark functions \(f_{1},f_{2},f_{5},f_{7},f_{8},f_{9},f_{10},f_{12}\,\text{and}\,f_{13}\), and that there is no significant difference between the two methods on the rest of the benchmark functions. Similarly, Wilcoxon signed-rank test shows that ICACI statistically significantly outperforms the Perturbed ICA on benchmark functions \(f_{1}\), \(f_{7}\), \(f_{8}\), \(f_{12}\) and \(f_{13}\), and that the Perturbed ICA statistically significantly \begin{table}
\begin{tabular}{c c c} \hline \(\mathbf{f}\) & **Range** & \(\mathbf{f_{\text{min}}}\) \\ \hline \(f_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & \(x_{i}\in[-100,100]\) & \(f_{1}(\mathbf{0})=0\) \\ \(f_{2}(x)=\sum_{i=1}^{n}\abs{x_{i}}+\prod_{i=1}^{n}\abs{x_{i}}\) & \(x_{i}\in[-10,10]\) & \(f_{2}(\mathbf{0})=0\) \\ \(f_{3}(x)=\sum_{i=1}^{n}\abs{\sum_{j=1}^{n}x_{j}^{2}}\) & \(x_{i}\in[-100,100]\) & \(f_{3}(\mathbf{0})=0\) \\ \(f_{4}(x)=\max\limits_{x}\abs{x_{i}}\abs{1\leq i\leq n}\) & \(x_{i}\in[-100,100]\) & \(f_{4}(\mathbf{0})=0\) \\ \(f_{5}(x)=\sum_{i=1}^{n}\abs{x_{i}+0.5}\abs{x_{i}}\) & \(x_{i}\in[-100,100]\) & \(f_{5}(\mathbf{p})=0\), \(-0.5\leq p_{i}<0.5\) \\ \(f_{6}(x)=-\sum_{i=1}^{n}x_{i}\sin(\sqrt{\abs{x_{i}}})\) & \(x_{i}\in[-500,500]\) & \(f_{6}(\mathbf{420.97})=-418.9829n\) \\ \(f_{7}(x)=\sum_{i=1}^{n-1}(100\times(x_{i=1}-x_{i}^{2})^{2}+(x_{i}-1)^{2})\) & \(x_{i}\in[-100,100]\) & \(f_{7}(\mathbf{1})=0\) \\ \(f_{8}(x)=10n+\sum_{i=1}^{n}(x_{i}^{2}-10\cos(2\pi\varsigma))\) & \(x_{i}\in[-10,10]\) & \(f_{8}(\mathbf{0})=0\) \\ \(f_{9}(x)=1+\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}\cos(\frac{x_{ i}}{\sqrt{i}})\) & \(x_{i}\in[-600,600]\) & \(f_{9}(\mathbf{0})=0\) \\ \(f_{10}(x)=20+e-20e^{-\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}}-e^{\frac{1}{n}x_{i}^{2 }\cos(2\pi\varsigma)}\) & \(x_{i}\in[-32,32]\) & \(f_{10}(\mathbf{0})=0\) \\ \(f_{11}(x)=-\sum_{i=1}^{n}(x_{i})\abs{\sin\frac{ix_{i}^{2}}{\pi}}\) & \(x_{i}\in[0,\pi]\) & \(f_{11}>-n\) \\ \(f_{12}(x)=\frac{\pi}{n}\{10\sin^{2}(\pi\varsigma_{1})+\sum_{i=1}^{n-1}(y_{i}-1 )^{2}[1+10\sin^{2}(\pi\varsigma_{i+1})]\) & \(x_{i}\in[-50,50]\) & \(f_{12}(\mathbf{-1})=0\) \\ \((y_{*}-1)^{2}\}+\sum_{i=1}^{n}\mu(x_{i},10,100,4)\), where \(y_{i}=1+(x_{i}+1)/4\) & \\ \(f_{13}(x)=0.1\{\sin^{2}(3\pi\varsigma_{i})+\sum_{i=1}^{n-1}(x_{i}-1)^{2}[1+\sin^{ 2}(3\pi\varsigma_{i+1})]\) & \(x_{i}\in[-50,50]\) & \(f_{13}(\mathbf{1})=0\) \\ \(+(x_{*}-1)^{2}[1+\sin^{2}(2\pi\varsigma_{*})]\}+\sum_{i=1}^{n}\mu(x_{i},5,100,4)\) & & \\ \hline \end{tabular}
\end{table} 
Table 1: Benchmark functions. outperforms ICACI only on benchmark function \(f_{9}\). Therefore, the addition of an interaction step in ICAAI and ICACI improves the Perturbed ICA on some benchmark functions. Wilcoxon signed-rank test between ICAAI and ICACI shows that ICAAI statistically significantly outperforms ICACI on benchmark functions \(f_{1}\), \(f_{4}\), \(f_{5}\), \(f_{7}\), \(f_{8}\), \(f_{9}\), \(f_{10}\), \(f_{12}\) and \(f_{13}\), and that there is no significant difference between the two methods on the rest of the benchmark functions. Therefore, the interaction step using artificial imperialist appears to be more effective than that using crossover imperialists.