# Crossover Rate Sorting in Adaptive Differential Evolution
## 3 Results
### Numerical Results
The results indicate that the effect of the sorting crossover rates is not always positive. For example, for L-NTADE, the sorting results in worse performance in most cases, although for \(100*D\) functions, there are 4 improvements out of 30. However, the NL-SHADE-LBC algorithm receives significant improvements, as well as L-SHADE-RSP, which becomes better for 7 functions out of 30. At the same time, the NL-SHADE-RSP algorithm does not seem to be influenced by this modification, and its performance stays almost at the same level. The Friedman ranking demonstrates that the algorithms with crossover rate sorting received lower or equal ranks in almost all cases, except L-NTADE in \(50*D\) and \(100*D\). Table 3 shows the comparison on the CEC 2022 benchmark, and Table 4 contains the Friedman ranking of the results. The results of the modified algorithms with crossover rate sorting, shown in Table 3, are much better, with the only exception being the NL-SHADE-RSP algorithm in the \(20*D\) case. Other algorithms achieved improvements on up to 6 functions out of 12, often without performance losses. However, it is important to mention here that most of the improvements are observed on functions, which are easily solved by the algorithm, and the improvement is in terms of convergence speed, i.e., number of function evaluations required to achieve the optimum. For the NL-SHADE-RSP algorithm, there was a performance loss observed on the 10-th function, i.e., Composition Function 2, combining Rotated Schwefel's Function, Rotated Rastrigin's Function and the HGBat Function. The [PERSON] ranking in Table 4 shows that the crossover rate sorting leads to significant performance improvements in all cases. The presented comparison shows that there is a certain effect of the crossover rate sorting; however, it is important to understand the reasons for the algorithms' different \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mathbf{10*D}\) & \(\mathbf{20*D}\) & **Total** \\ \hline LSHADE-RSP & 47.25 & 41.40 & 88.65 \\ LSHADE-RSP\_s & 41.83 & 34.45 & 76.28 \\ NL-SHADE-RSP & 54.77 & 59.25 & 114.02 \\ NL-SHADE-RSP\_s & 51.23 & 57.50 & 108.73 \\ NL-SHADE-LBC & 30.62 & 38.78 & 69.40 \\ NL-SHADE-LBC\_s & 23.63 & 30.33 & 53.97 \\ L-NTADE & 45.73 & 38.65 & 84.38 \\ L-NTADE\_s & 40.93 & 35.63 & 76.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Algorithms with and without crossover rate sorting, CEC 2022, [PERSON] ranks. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(\mathbf{10*D}\) & \(\mathbf{20*D}\) \\ \hline L-SHADE-RSP vs. & 4/8/0 & 4/8/0 \\ L-SHADE-RSP\_s & (27.44) & (25.92) \\ \hline NL-SHADE-RSP vs. & 6/5/1 & 2/9/1 \\ NL-SHADE-RSP\_s & (26.49) & (11.32) \\ \hline NL-SHADE-LBC vs. & 6/6/0 & 5/7/0 \\ NL-SHADE-LBC\_s & (36.62) & (31.43) \\ \hline L-NTADE vs. & 5/7/0 & 3/8/1 \\ L-NTADE\_s & (27.05) & (14.40) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Algorithms with and without crossover rate sorting, CEC 2022, Mann–Whitney tests and total standard score. behaviors and convergence speeds. For this purpose, first of all, the diversity measure is introduced as the average pairwise distance (\(APD\)) between individuals:  
\[APD=\frac{2}{N(N-1)}\sum_{i=1}^{N}\sum_{k=i+1}^{N}\sqrt{\sum_{j=1}^{D}(x_{i,j}-x _{k,j})^{2}} \tag{9}\]  
Figure 2 shows the diversity of the newest population in the L-NTADE algorithm with and without crossover rate sorting (CRS), CEC 2022, \(20*D\) case. The graphs in Figure 2 demonstrate that applying crossover rate sorting results in earlier diversity loss in the population in most cases. The only exception here is F4, Shifted and Rotated Non-Continuous Rastrigin's Function, where the diversity stays at a high level for a longer period. Hence, the improvement on the CEC 2022 benchmark can be explained by faster convergence. However, this convergence may be premature, for example, like on F10, where applying crossover rate sorting leads to early stop of the search process.