## 4 Experimental Results
### Performance Comparison
#### 4.2.1 Experiment 1: Impact of Interaction Operation
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\boldsymbol{F}\) & **PSO** & **Perturbed ICA [35]** & **ICAAI** & **ICACI** \\ \hline \(f_{1}\) & 1.57 \(\times\)\(10^{-16}\) (8.6 \(\times\)\(10^{-16}\)) & 8.312 \(\times\)\(10^{-6}\) (1.3 \(\times\)\(10^{-5}\)) & 3.757 \(\times\)\(10^{-10}\) (2 \(\times\)\(10^{-9}\)) & 2.1 \(\times\)\(10^{-7}\) (9.7 \(\times\)\(10^{-7}\)) \\ \(f_{2}\) & 1.015 \(\times\)\(10^{-3}\) (5.6 \(\times\)\(10^{-3}\)) & 3.559 \(\times\)\(10^{-4}\) (7.48 \(\times\)\(10^{-4}\)) & 1.103 \(\times\)\(10^{-7}\) (2.38 \(\times\)\(10^{-7}\)) & 5.08 \(\times\)\(10^{-5}\) (2.1 \(\times\)\(10^{-4}\)) \\ \(f_{3}\) & 1.5 \(\times\)\(10^{-18}\) (5.8 \(\times\)\(10^{-18}\)) & 2.687 \(\times\)\(10^{-4}\) (4.5 \(\times\)\(10^{-4}\)) & 1.53 \(\times\)\(10^{-10}\) (6.4 \(\times\)\(10^{-10}\)) & 2.835 \(\times\)\(10^{-6}\) (7 \(\times\)\(10^{-6}\)) \\ \(f_{4}\) & 14.44 (3.7) & 6.607 (2.2) & 1.989 \(\times\)\(10^{-1}\) (0.2) & 8.134 (3.1) \\ \(f_{5}\) & 6.67 \(\times\)\(10^{-2}\) (0.25) & 19.57 (37.7) & 0.3 (0.79) & 46.27 (146.88) \\ \(f_{6}\) & \(-\)1.135 \(\times\)\(10^{4}\) (367) & \(-\)1.140 \(\times\)\(10^{4}\) (280) & \(-\)1.142 \(\times\)\(10^{4}\) (256) & \(-\)1.143 \(\times\)\(10^{4}\) (304) \\ \(f_{7}\) & 31.2 (16.3) & 230.3 (295.9) & 100.2 (131.3) & 126.7 (145) \\ \(f_{8}\) & 38.5 (10.14) & 5.945 (3.03) & 5.172 (2.94) & 6.008 (2.89) \\ \(f_{9}\) & 1.53 \(\times\)\(10^{-2}\) (0.02) & 2.284 \(\times\)\(10^{-2}\) (0.03) & 1.23 \(\times\)\(10^{-2}\) (0.017) & 3.81 \(\times\)\(10^{-2}\) (0.037) \\ \(f_{10}\) & 8.4 \(\times\)\(10^{-7}\) (4.55 \(\times\)\(10^{-6}\)) & 1.203 \(\times\)\(10^{-3}\) (1.3 \(\times\)\(10^{-3}\)) & 4.139 \(\times\)\(10^{-6}\) (8.6 \(\times\)\(10^{-6}\)) & 1.063 \(\times\)\(10^{-3}\) (0.002) \\ \(f_{11}\) & \(-\)23.757 (1.22) & \(-\)27.72 (0.59) & \(-\)27.68 (0.89) & \(-\)27.58 (0.89) \\ \(f_{12}\) & 1.037 \(\times\)\(10^{-2}\) (0.03) & 6.913 \(\times\)\(10^{-3}\) (0.026) & 1.037 \(\times\)\(10^{-2}\) (0.032) & 6.91 \(\times\)\(10^{-3}\) (0.026) \\ \(f_{13}\) & 1.1 \(\times\)\(10^{-3}\) (3.35 \(\times\)\(10^{-3}\)) & 1.810 \(\times\)\(10^{-3}\) (4.04 \(\times\)\(10^{-3}\)) & 1.83 \(\times\)\(10^{-3}\) (4.16 \(\times\)\(10^{-3}\)) & 1.83 \(\times\)\(10^{-3}\) (4.16 \(\times\)\(10^{-3}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average and standard deviation of the best objective values over 30 runs after 1000 generations. The standard deviation is shown in parentheses. ICA on benchmark functions \(f_{1},f_{5},f_{7},f_{8},f_{9},f_{10}\) and \(f_{12}\), and there is no significant difference between ICAAI and the Perturbed ICA on the rest of the benchmark functions. ICAAI is still more effective than ICACI after 3000 generations. According to Wilcoxon signed-rank test, ICAAI statistically significantly outperforms ICACI on benchmark functions \(f_{1},f_{4},f_{5}\), \(f_{7}\), \(f_{8},f_{9}\) and \(f_{10}\), and there is no significant difference between ICAAI and ICACI on the rest of the benchmark functions. Both ICAAI and ICACI are more competitive to PSO after 3000 generations than after 1000 generations. According to Wilcoxon signed-rank test, ICAAI statistically significantly outperforms PSO on benchmark functions \(f_{3}\), \(f_{5}\), \(f_{6}\), \(f_{7}\) and \(f_{9}\), and PSO statistically significantly outperforms ICAAI on benchmark functions \(f_{1},f_{8},f_{10},f_{12}\) and \(f_{13}\). Similarly, Wilcoxon signed-rank test shows that ICACI statistically significantly outperforms PSO on benchmark functions \(f_{3},f_{6}\), \(f_{7}\) and \(f_{9}\), and that PSO statistically significantly outperforms ICACI on benchmark functions \(f_{1},f_{4},f_{5},f_{8},f_{10},f_{12}\) and \(f_{13}\).