# Crossover Rate Sorting in Adaptive Differential Evolution
## 3 Results
### Numerical Results
Two other important cases are hybrid functions F7 and F8, where diversity oscillated, but here, it is also lower with CRS. Similar trends are observed for other algorithms in both CEC 2022 and CEC 2017 benchmarks. Another question worth investigating is the influence of crossover rate sorting on the process of parameter adaptation. For this purpose, the average values of all memory cells were calculated for \(F\) and \(Cr\) for all algorithms. Figures 3-6 show the adaptation process on CEC 2017, \(50*D\) case. Figure 2: Average pairwise distance between individuals in the newest population, L-NTADE, CEC 2022, \(20*D\). Figure 4: Parameter adaptation with and without crossover rate sorting, NL-SHADE-RSP, CEC 2017, \(50*D\). Figure 3: Parameter adaptation with and without crossover rate sorting, L-SHADE-RSP, CEC 2017, \(50*D\). Considering the results in Figures 3-6, it can be observed that applying CRS results in larger \(Cr\) values stored in the memory cells in the majority of cases. Moreover, \(Cr\) often reaches 1, and rarely decreases, especially for L-SHADE-RSP and L-NTADE, where the memory size is rather small. The NL-SHADE-RSP is characterized by a more rapid increase in the average of the \(Cr\) values in the memory cells, while L-SHADE-LBC keeps larger \(Cr\) values for a longer period of time. The scaling factor parameter \(F\) is also affected, especially for the L-NTADE algorithm, where its behavior may differ significantly, e.g., in cases such as F8, F17, and F21. One explanation for this could be that decreased diversity leads to different \(F\) values to become more promising, usually resulting in larger \(F\) values in the memory cells. The presented graphs may lead to the following question: is the faster convergence and larger \(Cr\) values generated the only effects of the crossover rate sorting? Is it possible  
Figure 5: Parameter adaptation with and without crossover rate sorting, NL-SHADE-LBC, CEC 2017, \(50*D\). Figure 6: Parameter adaptation with and without crossover rate sorting, L-NTADE, CEC 2017, \(50*D\). to achieve the same results simply by setting larger \(Cr\) values, without any sorting? To answer these questions, another set of experiments was performed. In these experiments, the biased parameter adaptation approach [23] was applied; in particular, the Lehmer mean parameter \(p\) in memory cells update was increased. The standard setting for this parameter in L-SHADE-RSP, NL-SHADE-RSP and L-NTADE is 2, and it was increased to 4. In L-SHADE-LBC, this parameter is linearly controlled so that it is increased by 2. Larger values of \(p\) lead to a bias toward larger parameter values generated. Tables 5 and 6 show the comparison of the four used algorithm with the increased Lehmer mean parameter (with _lm in the name) and with crossover rate sorting on CEC 2017 and CEC 2022, respectively. Comparing Tables 1 and 5, it can be noted that in terms of number of wins/ties/losses, applying the biased Lehmer mean does not change the situation in most cases. However, the performance of NL-SHADE-RSP in high-dimensional problems is increased, and the algorithm with crossover rate sorting becomes much worse, especially in \(100*D\). Similar trends are observed for CEC 2022 in Table reftab6, where a significant difference may be observed for NL-SHADE-RSP in the \(10*D\) case--here adding more biased mean calculation leads to more similar performance to NL-SHADE-RSP_s, in terms of convergence speed. Table 7 shows the comparison of algorithms with crossover rate sorting with some of the top-performing algorithms, such as EBOwithCMAR [36] (first place in 2017), jSO [7] (second place) and LSHADE-SPACMA [37] (fourth place) on the CEC 2017 benchmark. Table 8 contains the comparison on the CEC 2022 benchmark with EAteigN100_10 [38] (first place in competition), NL-SHADE-RSP-MID [39] (third place in competition) and APGSK-IMODE [40].