## 4 Experimental Results
### Performance Comparison
#### 4.2.2 Experiment 2: Impact of Competition Frequency  
This experiment compares ICAAI and ICACI under various competition frequencies, which are controlled by the value of parameter \(\rho\). Figure 6 shows that ICAAI consistently outperforms ICACI on all of the five unimodal benchmark functions \(f_{1}\)\(\sim\)\(f_{5}\). For \(f_{1}\)\(\sim\)\(f_{4}\), ICAAI often yields better results with smaller competition frequency (i.e., smaller \(\rho\)). Figure 7 shows the results for the eight multimodal benchmark functions \(f_{6}\)\(\sim\)\(f_{13}\). ICAAI consistently outperforms ICACI on \(f_{9}\) and \(f_{10}\). For \(f_{7}\), \(f_{8}\) and \(f_{11}\), ICAAI underperforms ICACI only when \(\rho=0\). For \(f_{6}\) and \(f_{12}\), ICAAI often yields better results than ICACI does. Overall, ICACI performs better than ICAAI only on \(f_{13}\). Smaller competition frequency does not guarantee better performance for both ICAAI and ICACI, however, experimenting with \(\rho<1\) can often find better solutions. For example, Figure 6 and Figure 7 show that the best \(\rho\) for ICAAI is always less than 1 on all benchmark functions. \begin{table}
\begin{tabular}{l r r r r} \hline \hline \(f\) & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**Perturbed ICA [35]**} & \multicolumn{1}{c}{**ICAAI**} & \multicolumn{1}{c}{**ICACI**} \\ \hline \(f_{1}\) & \(1.89\times 10^{-54}\) (\(9.95\times 10^{-54}\)) & \(8.199\times 10^{-24}\) (\(3.3\times 10^{-23}\)) & \(2.89\times 10^{-28}\) (\(1.58\times 10^{-27}\)) & \(2.23\times 10^{-26}\) (\(9.5\times 10^{-26}\)) \\ \(f_{2}\) & \(1.015\times 10^{-3}\) (\(5.56\times 10^{-3}\)) & \(1.726\times 10^{-15}\) (\(3.33\times 10^{-15}\)) & \(2.671\times 10^{-19}\) (\(6.1\times 10^{-19}\)) & \(9.29\times 10^{-18}\) (\(2.2\times 10^{-17}\)) \\ \(f_{3}\) & \(3.17\times 10^{-56}\) (\(1.5\times 10^{-55}\)) & \(1.493\times 10^{-22}\) (\(4.56\times 10^{-22}\)) & \(1.665\times 10^{-25}\) (\(9.1\times 10^{-25}\)) & \(1.1\times 10^{-24}\) (\(3.6\times 10^{-24}\)) \\ \(f_{4}\) & \(6.67\) (\(2.77\)) & \(2.770\times 10^{-1}\) (\(0.19\)) & \(5.242\times 10^{-3}\) (\(6.17\times 10^{-3}\)) & \(3.396\times 10^{-1}\) (\(0.2\)) \\ \(f_{5}\) & \(0\) (\(0\)) & \(19.57\) (\(37.7\)) & \(0.3\) (\(0.79\)) & \(46.1\) (\(146.9\)) \\ \(f_{6}\) & \(-1.136\times 10^{4}\) (\(368.7\)) & \(-1.14\times 10^{4}\) (\(280.32\)) & \(-1.142\times 10^{4}\) (\(263\)) & \(-1.143\times 10^{4}\) (\(303.6\)) \\ \(f_{7}\) & \(29.96\) (\(16.19\)) & \(87.34\) (\(118.4\)) & \(49.39\) (\(59.02\)) & \(59.07\) (\(84.66\)) \\ \(f_{8}\) & \(28.73\) (\(10.34\)) & \(3.681\) (\(2.22\)) & \(4.676\) (\(3.02\)) & \(5.373\) (\(3.44\)) \\ \(f_{9}\) & \(1.53\times 10^{-2}\) (\(0.02\)) & \(2.282\times 10^{-2}\) (\(0.031\)) & \(1.230\times 10^{-2}\) (\(0.017\)) & \(3.807\times 10^{-2}\) (\(0.037\)) \\ \(f_{10}\) & \(1.8\times 10^{-14}\) (\(4.2\times 10^{-15}\)) & \(6.05\times 10^{-13}\) (\(1.5\times 10^{-12}\)) & \(1.68\times 10^{-13}\) (\(3.22\times 10^{-13}\)) & \(8.79\times 10^{-13}\) (\(1.8\times 10^{-12}\)) \\ \(f_{11}\) & \(-24.42\) (\(1.16\)) & \(-28.02\) (\(0.74\)) & \(-27.93\) (\(0.84\)) & \(-27.59\) (\(0.91\)) \\ \(f_{12}\) & \(6.9\times 10^{-3}\) (\(0.026\)) & \(6.911\times 10^{-3}\) (\(0.026\)) & \(1.037\times 10^{-2}\) (\(0.0317\)) & \(6.911\times 10^{-3}\) (\(0.0263\)) \\ \(f_{13}\) & \(7.325\times 10^{-4}\) (\(0.0028\)) & \(1.099\times 10^{-3}\) (\(3.35\times 10^{-3}\)) & \(1.831\times 10^{-3}\) (\(4.17\times 10^{-3}\)) & \(1.831\times 10^{-3}\) (\(4.16\times 10^{-3}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average and standard deviation of the best objective values over 30 runs after 300 generations. The standard deviation is shown in parentheses. Figure 6: ICAAI (solid line) and ICACI (dashed line) for unimodal benchmark functions \(f_{1}\)-\(f_{5}\). The horizontal axis is \(\rho\), and the vertical axis is the average of the best objective values. Figure 7: ICAAI (solid line) and ICACI (dashed line) for multimodal benchmark functions \(f_{6}\)-\(f_{13}\).