# Multi-Population Differential Evolution Algorithm with Uniform Local Search
## 5 Experimental Results
### Experimental Results and Comparisons with Other DE Variants  
In this subsection, the advantage of MUDE is obtained by extensively comparing MUDE with four other up-to-date DE versions, CoDE, JDE, JADE, and SaDE, whose control parameters \(F\) and \(CR\) are consistent with the original literature. The maximum number of fitness evaluations (MaxFES) is fixed at 3 \(\times\) 10\({}^{5}\) in all algorithms. Table 1 shows the experimental comparison results of 25 runs on 25 functions. The [PERSON] test results are shown at the bottom of Table 1, where the symbol \"\(-/+/\approx\)\" represents whether the average value of the algorithm is worse, better, or similar to that of MUDE.  
Table 2 shows the mean and standard deviation of the optimal fitness for 25 independent runs, and the best mean values are shown in bold. It is seen that MUDE is better than CoDE, JADE, JDE, and SaDE on 12, 11, 14, and 14 functions, respectively, but worse than these challengers on 8, 6, 3, and 5 functions, respectively. For the unimodal functions \(F_{1}\)-\(F_{5}\), the MUDE performance is superior to the other four algorithms for this type of benchmark function. As for the multimodal function \(F_{6}\)-\(F_{25}\), MUDE outperforms CoDE, JADE, JDE, and SaDE on 8, 9, 10, and 10 functions, respectively. CoDE beats MUDE on \(F_{6}\), \(F_{8}\), \(F_{11}\), \(F_{14}\), and \(F_{22}\). JADE exceeds MUDE on \(F_{5}\), \(F_{13}\), and \(F_{14}\). jDE and SaDE excel MUDE in only one function. Accordingly, MUDE's performance is the best one in 20 multimodal functions. It is clear that information migration of MUDE can improve population diversity.  
Along with the prior analysis, [PERSON]'s test and [PERSON]'s signed-rank test were performed on all function experimental results. The average ranking of five algorithms is provided in Table 3. [PERSON]'s statistic result shows that the smaller the value, the better the performance. Therefore, MUDE is the most competitive one in the five DE variants algorithms.  
The statistical results indicate that MUDE is the most challenging algorithm for 25 benchmark functions. There are two aspects reasons why MUDE can perform well. First, the ULS of MUDE leads to efficient exploitation and high convergence precision under such circumstances. Second, the information migrating mechanism among different islands improves population diversity.  
To illustrate the convergence of algorithm, fitness variation graphs of the 25 functions were made for the convergence course, which includes the mean fitness values over 25 runs with dimension \(D=30\). Figure 4 shows the convergence graphs of the 25 functions. In Figure 4, MUDE can continuously converge to better solutions for the unimodal function, but for the multimodal function, the algorithm may get stuck in a local solution. For function \(F_{18}\)-\(F_{24}\), convergence curves are almost the same. In general, MUDE improves the convergence of the DE algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**CoDE**} & \multicolumn{2}{c}{**JADE**} & \multicolumn{2}{c}{**JDE**} & \multicolumn{2}{c}{**SADE**} & \multicolumn{2}{c}{**MDDE**} \\ \hline \(F\) & **Mean** & **Ssd** & **Mean** & **Ssd** & **Mean** & **Ssd** & **Mean** & **Ssd** & **Mean** & **Ssd** \\ \hline \(F_{1}  
Furthermore, the total running time of the 25 functions is compared in five different DE algorithms, as shown in Figure 5.  
In Figure 5, the abscissa and ordinate represent the running time (h) and different DE algorithms, respectively. The results show that JADE has the shortest running time and jDE has the longest running time. The proposed MODE leads to a longer running time because of population migration and local search.