# Parameter Combination Framework for the Differential Evolution Algorithm
## 6 Experiments and Discussion
Since the current evolutionary operators are not very effective for these rotation functions, there lacks a wide range of suitable parameter values. After a period, the parameter \(CR\) is still locked to 0 for a one-dimensional search in SHADE. In this type of case, the parameter-value combination scheme loses its effect in the later stage, thus affecting its efficiency. f30 is another case in which both algorithms are apt to local extremum; however, PVCDE has more opportunities to fall into a larger local extremum. For other functions, such as f1, f5, f6, f7, f12, f14, f17, f18, f20, and f21, PVCDE is better than SHADE on most high-dimensional issues, which demonstrates the effect of the parameter combination strategy. For example, so\({}_{1}\) was the most useful for f1, and for f6 in 30\(D\), the combination of so\({}_{1}\) and so\({}_{2}\) assisted so\({}_{3}\) to reduce the cases involving falling into a local extremum. The process of parameter distribution variation is demonstrated in Figure 5. The sample function is f12 (50\(D\)). The figure was intercepted during the runtime in different generations. The red stars in the figure represent the successful parameters which make the individuals evolve successfully. We can observe the variation process. In the initial stages, the region 1 is strengthened by so\({}_{1}\), and region 4 is also give a little percentage by so\({}_{2}\). Then the success rate of region 1 is decreased and the evolution enters the stage2. The three search objects competed. The region 2 get higher density. At later period, so\({}_{1}\) and so\({}_{2}\)'s CR is locked to 0, but so\({}_{3}\) which does not use this technology moved to left boundary. The strategy of PVCDE can bring more control ability to parameter distribution. In the second experiment, we combined the second category parameter strategy with our method: we added the parameter strategy LPSR for the \(NP\) into our framework. LPSR is a strategy proposed by L-SHADE that reduces the population size linearly according to the number of function evaluations. L-SHADE is a variant of SHADE with LPSR. Because L-SHADE was successful in the CEC2014 competition, we constructed L-PVCDE with the same \(NP\) strategy and compared to it. The algorithm settings are as follows:  
* L-SHADE: the parameter settings is same as its initial settings in paper [18]. The population size is from 18\(D\) to 4, and the parameters for DE/current-to-pbest/1 operation are set to \(p\) = 0.11, archive size = 2.6*\(NP\), and memory size = 6. * L-PVCDE: the population size is from 18\(D\) to 10 for the grouping restriction. The initial group size ratios (Igsr) are set as [16/18, 1/10, 1/10], and the basic proportions of all search objects in stage 1 are set to [15/18, 1/18, 1/18]. The other settings are the same as PVCDE. We performed the test on \(D\) = 50 dimensions, with 25 runs for 30 functions. The test results are shown in Table 5. The mean and standard deviation of each algorithm are listed in the Table 5. Wilcoxon rank-sum test was applied to compare L-SHADE and L-PVCDE. The symbols \(-\), \(+\), and \(\approx\) represent that the L-SHADE algorithm is significantly worse than, better than, or similar to L-PVCDE, respectively. Since L-PVCDE uses the L-SHADE strategy only with a combination of parameter regions, L-PVCDE is compared to L-SHADE to test if the combination is useful. Figure 5: The process of parameter distribution (F*CR) variation for f12 (50D). As seen in Table 5, L-PVCDE obtains better results than L-SHADE with 16 functions, and worse results than L-SHADE with seven functions. There functions for which L-PVCDE is worse than L-SHADE including f9, f10, f13, f15, f16, f26, and f23 according to the result of Wilcoxon rank-sum test. Whereas, there are some functions for which L-PVCDE is superior to L-SHADE.