# Crossover Rate Sorting in Adaptive Differential Evolution
## 4 Discussion  
The results presented in the previous section show that applying sorting of the crossover rate values has a significant effect on the diversity of the population, convergence speed and even parameter adaptation of modern adaptive differential evolution. The reason for the lower diversity is that assigning larger crossover rate parameter values \(Cr\) to worse individuals leads to potentially larger changes in them. In other words, worse individuals are changed more significantly and hence are faster driven toward better objective function values. This, in turn, has two major effects. Firstly, the diversity is decreased faster, and secondly, larger crossover rate values receive higher improvements, resulting in larger \(Cr\) in memory cells. The second effect also leads to larger \(Cr\) values being assigned also to the best individuals in the population, compared to the same algorithm without crossover rate sorting.  
All the described effects of the sorting \(Cr\) values are not always positive and depend on the algorithm used. For example, in the performed experiments on the CEC 2017 benchmark, where the computational resource is rather small, dimensions are high, and problems are diverse, applying sorting in L-SHADE-RSP and L-SHADE-LBC gives a positive effect, while NL-SHADE-RSP and L-NTADE show decreased performance as the dimension increases. In the case of the CEC 2022 benchmark, where the convergence speed is important, sorting \(Cr\) results in increased performance for all algorithms. An additional experiment with increased bias in parameter adaptation showed that sorting is not the same as simply increasing the generated \(Cr\) values, and it has a more complicated effect on the differential evolution performance. In general, a similar sorting could be applied to other parameters and algorithms. However, the preliminary experiments with scaling factor \(F\) sorting did not show any performance improvements, even though the tests were  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & \(\mathbf{10*D}\) & \(\mathbf{20*D}\) \\ \hline LSHADE-RSP\_s vs. & \(2/3/7\) & \(5/3/4\) \\ EA4 eigN100\_10 & \((-36.18)\) & \((5.54)\) \\ \hline LSHADE-RSP\_s vs. & \(4/2/6\) & \(6/2/4\) \\ NL-SHADE-RSP-MID & \((-7.53)\) & \((16.98)\) \\ \hline LSHADE-RSP\_s vs. & \(7/1/4\) & \(9/1/2\) \\ APGSK-IMODE & \((28.48)\) & \((49.20)\) \\ \hline NL-SHADE-RSP\_s vs. & \(1/2/9\) & \(2/1/9\) \\ EA4 eigN100\_10 & \((-54.42)\) & \((-51.07)\) \\ \hline NL-SHADE-RSP\_s vs. & \(2/4/6\) & \(1/5/6\) \\ NL-SHADE-RSP-MID & \((-27.32)\) & \((-27.73)\) \\ \hline NL-SHADE-RSP\_s vs. & \(7/2/3\) & \(5/2/5\) \\ APGSK-IMODE & \((18.97)\) & \((3.82)\) \\ \hline NL-SHADE-LBC\_s vs. & \(5/3/4\) & \(5/3/4\) \\ EA4 eigN100\_10 & \((7.96)\) & \((8.30)\) \\ \hline NL-SHADE-LBC\_s vs. & \(5/4/3\) & \(6/2/4\) \\ NL-SHADE-RSP-MID & \((15.33)\) & \((10.70)\) \\ \hline NL-SHADE-LBC\_s vs. & \(10/1/1\) & \(8/1/3\) \\ APGSK-IMODE & \((58.57)\) & \((38.79)\) \\ \hline L-NTADE\_s vs. & \(2/4/6\) & \(3/5/4\) \\ EA4 eigN100\_10 & \((-28.38)\) & \((-3.66)\) \\ \hline L-NTADE\_s vs. & \(4/4/4\) & \(7/1/4\) \\ NL-SHADE-RSP-MID & \((-3.38)\) & \((16.63)\) \\ \hline L-NTADE\_s vs. & \(7/3/2\) & \(10/0/2\) \\ APGSK-IMODE & \((37.75)\) & \((51.52)\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Algorithms with crossover rate sorting vs. alternative approaches, CEC 2022, Mann–Whitney tests and total standard score.  
performed with sorting in ascending and descending order. Nevertheless, some other approaches, such as genetic algorithms, which use parameter adaptation, may benefit from sorting generated parameter values, although aspect this requires additional studies.