## 4 Performance Analysis
### Analysis of Experimental Results  
To facilitate an intuitive understanding of TCDE, Figure 8 presents the ultimate outcomes achieved with TCDE for eight distinct functions. From Figure 8, it becomes evident that the TCDE algorithm effectively identifies all peak points within these eight functions. These functions encompass not only simpler ones, such as F4(a) and F5(b), but also more intricate functions like F6(c), F7(d), and F10(e), which possess numerous peak points. Furthermore, even in the case of highly complex functions such as F12(g) and F13(h), which contain many local optima, the TCDE algorithm consistently and accurately pinpoints all peaks with the guidance provided by the topographical contours. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Function** & **Population Size (N)** & **MaxFE** \\ \hline F4 & 80 & 5.00 \(\times\) 10\({}^{4}\) \\ F5 & 80 & 5.00 \(\times\) 10\({}^{4}\) \\ F6 & 100 & 2.00 \(\times\) 10\({}^{5}\) \\ F7 & 300 & 2.00 \(\times\) 10\({}^{5}\) \\ F10 & 100 & 2.00 \(\times\) 10\({}^{5}\) \\ F11 & 200 & 2.00 \(\times\) 10\({}^{5}\) \\ F12 & 200 & 2.00 \(\times\) 10\({}^{5}\) \\ F13 & 200 & 2.00 \(\times\) 10\({}^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Population size and maximum number of fitness evaluations [39]. Both the TCDE and AnDE algorithms perform well, but the former outperforms the latter (which uses contour methods) on F7 and F13. As shown in Figure 8, these two test functions involve complex scenarios with coexisting rough and fine peaks, as well as steep connections between peaks. This indicates that, compared with the AnDE algorithm's local application of contour optimization, the TCDE algorithm's strategy of globally optimizing topographical contours while distinguishing between rough and fine peaks is effective, enabling it to handle multimodal optimization problems in complex situations. As shown in Tables 3-5, all baseline methods in the subpopulation group perform worse than TCDE. Among them, Self-CSDE shows extremely poor performance: its PR and SR values are only 1.000 for the F5 function in Table 3, and it fails to achieve perfect results for any other test functions. This indicates that its niche strategy of adaptively generating subpopulations fails in complex scenarios with steep connections between multimodal peaks and coexisting rough-fine peaks. CDE, SDE, and PNPCDE can effectively handle scenarios with gentle or steep connections between multimodal peaks, but their  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c} \hline \hline Function & \multicolumn{2}{c}{TCDE} & \multicolumn{2}{c}{ANDE} & \multicolumn{2}{c}{CDE} & \multicolumn{2}{c}{SDE} & \multicolumn{2}{c}{LIPS} & \multicolumn{2}{c}{R3 PSO} & \multicolumn{2}{c}{NCD-DE} & RS-CMSA-ESII \\  & PR & SR & PR & SR & PR & SR & PR & SR & PR & SR & PR & SR & PR & SR & PR & SR \\ \hline F4 & **1.000** & **1.000** & **1.000** & 0.995 & 0.980 & 0.284 &  
strategies perform poorly in complex cases with rough-fine peaks. Although RS-CMSA-ESII demonstrates improved performance, it still has a probability of failure in dealing with steep connections between multimodal peaks and complex situations with coexisting rough-fine peaks. Similarly, it is evident that TCDE outperforms the baseline methods relying on local information. Here, methods such as LIPS, R3 PSO, and LoISDE, which primarily depend on local information strategies, struggle to locate all peaks in both scenarios with gentle connections between multimodal peaks and more complex ones. When local information is combined with strategies such as Gaussian distribution and fitness-distance enhancement, the multimodal optimization capabilities of the algorithms are enhanced. For instance, LMCEDA, LMSEDA, and FDLS-ADE can not only handle gentle connections between multimodal peaks but also demonstrate significantly improved performance in scenarios with steep connections.