# Knowledge-Based Perturbation LaF-CMA-ES for Multimodal Optimization
## 4 Experiment and Analysis
### Experimental Analysis and Result
These \(p\)-values are derived from the results, with the Bonferroni-Dunn method employed to compute adjusted \(p\)-values. The test results are presented in Table 6.  
Table 6 displays both the \(p\)-values and the adjusted \(p\)-values, highlighting differences among the five algorithms across all three dimensions. Specifically, in the 50D and 100D dimensions, statistical disparities have emerged between the KbP-LaF-CMAES and IPOP-CMAES algorithms. To confirm the directionality of these statistical differences, reference to the raw data is necessary. Further statistical analysis of the simulation results involves making an all-pair comparison without a control algorithm. The outcomes of the \(n\times n\) comparison using post hoc Friedman tests for the three kinds of dimensions are detailed in Tables 7-9, respectively. n is the count of the compared algorithms. From these tables, the \(\frac{n\times(n-1)}{2}\) pairwise comparison results can be derived. Specifically, for the 50D and 100D test functions, the introduced method outperforms the other compared algorithms. The experimental results demonstrate a significant advantage of the introduced method over the comparison algorithms. The statistical analysis indicates that the introduced method exhibits distinct performance improvements over SPSRDEMMS and the three other variants of CMA-ES, validating its efficiency. The superiority of the introduced method becomes more evident as dimensionality increases. The comparison between KbP-LaF-CMAES and EBOwithCMAR reveals that the performance of KbP-LaF-CMAES is comparable to that of the state-of-the-art algorithm. Detailed analysis of the experimental results shows that KbP-LaF-CMAES performs well on several typical multimodal functions, such as F7, F8, F14, F15, F18, F19, and F20. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **KbP-LaF-CMAES** & **EBOwithCMAR** & **IPOP-CMAES** & **LaF-CMAES** & **CMA-ES** & **SPSRDEMMS** \\ \hline KbP-LaF-CMAES & 0.972 & 0.093 & 0.046 & 0.042 & 0.001 \\ EBOwithCMAR & 0.972 & & 0.100 & 0.049 & 0.046 & 0.001 \\ IPOP-CMAES & 0.093 & 0.100 & & 0.748 & 0.721 & 0.116 \\ LaF-CMAES & 0.046 & 0.049 & 0.748 & & 0.972 & 0.211 \\ CMA-ES & 0.042 & 0.046 & 0.721 & 0.972 & & 0.225 \\ SPSRDEMMS & 0.001 & 0.001 & 0.116 & 0.211 & 0.225 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of 6 vs. 6 comparisons on the benchmark of 30D. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **KbP-LaF-CMAES** & **EBOwithCMAR** & **IPOP-CMAES** & **LaF-CMAES** & **CMA-ES** & **SPSRDEMMS** \\ \hline KbP-LaF-CMAES & 0.391 & 0.018 & 0.017 & 0.001 & 0.000 \\ EBOwithCMAR & 0.391 & & 0.694 & 0.069 & 0.007 & 0.002 \\ IPOP-CMAES & 0.018 & 0.694 & & 0.972 & 0.372 & 0.199 \\ LaF-CMAES & 0.017 & 0.069 & 0.972 & & 0.391 & 0.211 \\ CMA-ES & 0.001 & 0.007 & 0.372 & 0.391 & & 0.694 \\ SPSRDEMMS & 0.000 & 0.002 & 0.199 & 0.211 & 0.694 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of 6 vs. 6 comparisons on the benchmark of 50D. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **KbP-LaF-CMAES** & **EBOwithCMAR** & **IPOP-CMAES** & **LaF-CMAES** & **CMA-ES** & **SPSRDEMMS** \\ \hline KbP-LaF-CMAES & 0.568 & 0.038 & 0.024 & 0.004 & 0.000 \\ EBOwithCMAR & 0.568 & & 0.134 & 0.093 & 0.020 & 0.000 \\ IPOP-CMAES & 0.038 & 0.134 & & 0.858 & 0.411 & 0.001 \\ LaF-CMAES & 0.024 & 0.093 & 0.858 & & 0.520 & 0.002 \\ CMA-ES & 0.004 & 0.020 & 0.411 & 0.520 & & 0.017 \\ SPSRDEMMS & 0.000 & 0.000 & 0.001 & 0.002 & 0.017 & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of 6 vs. 6 comparisons on the benchmark of 100D. Additionally, KBP-LaF-CMAES also excels in the two composite problems F22 and F23. For the remaining multimodal and composite problems, KBP-LaF-CMAES exhibits comparable or slightly inferior performance compared to the other algorithms. Overall, the KBP-LaF-CMAES algorithm has demonstrated preferable performance on most test functions, particularly in high-dimensional problems.