# Knowledge-Based Perturbation LaF-CMA-ES for Multimodal Optimization
## 4 Experiment and Analysis
### Experimental Analysis and Result  
Simulations were conducted across three different dimensions. Five statistical indicators--Best, Worst, Median, Mean, and Standard Deviation (Std)--were calculated, as shown at the end of this paper. Each algorithm was run independently 51 times for each instance, and the Mean and Std indicators were derived from the performance of each algorithm on the test functions. The results of the introduced method on the CEC benchmark across the three dimensions are illustrated in Table 5, with the best performance for each function highlighted in bold. The test functions are notably complex due to their high dimensionality. Consequently, comparison experiments were performed on the test problems across these three types of dimensions to assess the robustness of the introduced algorithms. The statistical variances between the introduced method and the compared algorithms are analyzed using [PERSON]'s test. The results of [PERSON]'s test for the improved method and comparison algorithms across each dimension are illustrated in Table 6. These figures consistently show Kbp-LaF-CMAES achieving the highest ranking for the test benchmark. Subsequently, the significance levels of the differences between the algorithms are computed using [PERSON]'s test. \[CD=q_{\alpha}\sqrt{\frac{k(k+1)}{6N}} \tag{30}\]  
In Equation (30), \(k\) and \(N\) represent the number of compared algorithms and test functions, respectively, while \(\alpha\) denotes the confidence level, commonly set to 0.05 or 0.1. The critical difference for [PERSON]'s test was calculated with \(\alpha\) = 0.05 and \(\alpha\) = 0.1, as shown in Equation (30). In this study, with k = 5 and N = 28, \(q_{\alpha}\) = 2.394 for \(\alpha\) = 0.05, and \(q_{\alpha}\) = 2.128 for \(\alpha\) = 0.1. The statistical results are presented in Table 5, demonstrating that the introduced method outperforms the comparison methods. Regardless of the \(\alpha\) value and the dimension, a significant difference is detected among the comparison methods. The experimental results lead to several conclusions. The performance of the proposed algorithm surpasses that of most comparison algorithms across a wide range of functions in all dimensions. For low-dimensional problems (30D and 50D), the effectiveness of the introduced method is comparable to that of EBOwithCMAR. Subsequently, post hoc tests were conducted to identify the differences between these algorithms [38]. The introduced \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**30D**} & \multicolumn{2}{c}{**50D**} & \multicolumn{2}{c}{**100D**} \\  & \multicolumn{2}{c}{**Adjusted**} & \multicolumn{1}{c}{**\(p\)-Value**} & \multicolumn{1}{c}{**\(p\)-Value**} & \multicolumn{1}{c}{**\(p\)-Value**} & \multicolumn{1}{c}{**Adjusted**} \\  & \multicolumn{1}{c}{**\(p\)-Value**} & & & \multicolumn{1}{c}{**\(p\)-Value**} & & \multicolumn{1}{c}{**\(p\)-Value**} \\ \hline
**SPSRDEMMS** & 0.001 & 0.017 & 0.000 & 0.004 & 0.000 & 0.000 \\
**CMA-ES** & 0.042 & 0.627 & 0.001 & 0.017 & 0.004 & 0.057 \\
**LaF-CMAES** & 0.048 & 0.683 & 0.017 & 0.251 & 0.024 & 0.367 \\
**IPOP-CMAES** & 0.093 & 1.000 & 0.018 & 0.276 & 0.038 & 0.575 \\
**EBOwithCMAR** & 0.972 & 1.000 & 0.568 & 1.000 & 0.568 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the post hoc test. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Mean Rank (30D)** & **Mean Rank (50D)** & **Mean Rank (100D)** \\ \hline
**Kbp-LaF-CMAES** & **2.75** & **2.48** & **2.41** \\
**CMA-ES** & 3.77 & 4.11 & 3.86 \\
**LaF-CMA-ES** & 3.75 & 3.68 & 3.54 \\
**IPOP-CMA-ES** & 3.59 & 3.66 & 3.45 \\
**EBOwithCMAR** & 2.77 & 2.77 & 2.70 \\
**SPSRDEMMS** & 4.38 & 4.30 & 5.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: [PERSON]’s test rankings of the different algorithms. method served as the control algorithm for these tests. The objective of the post hoc tests is to establish a comparison set of \(p\)-values.