# Parameter Combination Framework for the Differential Evolution Algorithm
## 3 The Effect of Parameter Distribution on DE Algorithm
Premature convergence is identified by the state that the individual distribution is converged in a wrong region for a long period. As in individual distribution subgraph of Figure 2c, the first dimension is converged to 1e-5 precision, but the offset between the convergence location and the optimal value location is 1e1 precision, and has not improved in many generations. However, the jDE strategy that was apt to converge in the first test found the correct region and yielded optimal result. These two tests demonstrate that operation is related to parameter strategy. For the DE/rand/1 operation, the jDE strategy performs better than the SHADE strategy on test function \(\Omega\), while for  
Figure 1: Individual and parameters distribution maps of three parameter strategies for f1: (**a**) random strategy; (**b**) jDE strategy; and (**c**) SHADE strategy. Figure 2: Individual and parameter distribution maps of three parameter strategies for f2: (**a**) random strategy; (**b**) jDE strategy; and (**c**) SHADE strategy. the DE/current-to-pbest/1 operation, the jDE strategy performs more poorly than SHADE on test function f1. The SHADE strategy, which is generally an excellent parameter strategy, is not ideal for all operations. Thus, different combination mode of operation and parameter strategy should produce different search characteristics and may be suitable for different problems. Since jDE and SHADE parameter strategy produce very different parameter distribution characteristics, we consider that the difference of algorithm performance in the tests is due to the distribution of the two parameters \(F\) and \(CR\). We designed further experiments based on the second test to verify whether changing the parameter distribution would improve the performance of the algorithm. The algorithm is identical with the second test except the parameter strategy, and the test function is still \(\Omega\). For the parameter strategy we designed several different parameter region combinations. For the sake of discussion, we divided the domain of \(F\) and \(CR\) into two subdomains: large value [0.5, 1] and small value [0, 0.5]. Then, the plane of \(F\) and \(CR\) formed four regions, as shown in Figure 3. The experiment on all types of region combination patterns was performed, and the results are shown in Table 1. For each combination, the subregions are assigned the same number of individuals. Each region uses the SHADE parameter strategy. Premature convergence is identified as described in the second test. Initial domain is including region 1-4, but without division and combination. From Table 1, we can conclude that different regions have different effects and that combination patterns can affect the distribution of parameters and strengthen or weaken the effect. For example, with the region 4, premature convergence occurs, but when combined with the region 1, the result is superior than those for either region individually. The proper distribution or combination of parameter values can thus be useful for algorithms. From the experiments, we acquire several conclusions: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Region Combination Pattern** & **Result** & **Premature Convergence** \\ \hline (1) & 7.6e+02 & N \\ (2) & 1.0e+04 & N \\ (3) & 5.2e+03 & Y \\ (4) & 1.4e+03 & Y \\ **(1,4)** & **3.1e-03** & **N** \\ **(1,2)** & **6.0e+00** & **N** \\ (1,3) & 8.4e+02 & Y \\ (2,3) & 6.4e+02 & Y \\ (2,4) & 4.1e+03 & Y \\ (3,4) & 7.1e+02 & Y \\ **(1,2,4)** & **1.3e-01** & **N** \\ (1,3,4) & 2.0e+03 & Y \\ (1,2,3) & 3.5e+02 & N \\ (2,3,4) & 4.3e+03 & Y \\ (1,2,3,4) & 2.9e+02 & N \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results for all region combination patterns. Figure 3: Parameter region division. 1. different parameter control strategies may produce different parameter distribution. 2. the parameter distribution will influence the performance of the algorithm. 3. the parameter strategy is related to the operation adopted by DE variants. 4. adjusting the parameter distribution makes sense for improving the algorithm.