# Parameter Combination Framework for the Differential Evolution Algorithm
## 6 Experiments and Discussion  
In this section, we describe several experiments to test the proposed framework and methods. We performed an evaluation test on the CEC2014 Special Session on Real-Parameter Single Objective Optimization benchmark suite [40]. The CEC2014 benchmark set consists of 30 test functions. Functions F1-F3 are unimodal functions, F4-F16 are simple multimodal functions, F17-F22 are hybrid functions, and F23-F30 are composite functions. The search space is [\(-\)100, 100]\({}^{\text{D}}\). For every function, the maximum number of evaluations is 10,000 \(\times\)\(D\). According to the rules of CEC2014, the error between the best value found and the true optimal value is considered to be zero if it less than 1e-8. The algorithm in this paper is implemented with the MATLAB language and run in MATLAB R2018a. We present the first test between SHADE and PVCDE. SHADE is a superior DE variant. PVCDE uses the same operations as SHADE, and the parameter strategy of PVCDE in Level-2 is also similar to that of SHADE. The only difference between them is the region division and combination. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stage** & **Search Object** & **Parameter** & **Parameter Scope** & **Parameter Control Strategy** \\ \hline \multirow{4}{*}{exploration} & so\({}_{1}\) & \(F\) and \(CR\) & \(\{0.5,1\}\times[0.5,1]\) (Region\({}_{1}\)) & SHADE \\  & \(p\) & 0.1 & fixed \\ \cline{2-5}  & so\({}_{2}\) & \(F\) and \(CR\) & \(\{0.0,5\}\times[0.5,1]\) (Region\({}_{1}\)) & SHADE \\  & \(p\) & 0.1 & fixed \\ \cline{2-5}  & so\({}_{3}\) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Region\({}_{5}\)) & SHADE \\  & \(p\) & 0.5 & fixed \\ \hline \multirow{4}{*}{explotation} & so\({}_{1}\) & \(F\) and \(CR\) & \(\{0.5,1\}\times[0,1]\) (Region\({}_{1}\)) & SHADE \\  & \(p\) & 0.1 & fixed \\ \cline{2-5}  & so\({}_{2}\) & \(F\) and \(CR\) & \(\{0.0,5\}\times[0.0,5]\) (Region\({}_{1}\)) & SHADE \\  & \(p\) & 0.1 & fixed \\ \cline{2-5}  & so\({}_{3}\) & \(F\) and \(CR\) & \(\{0.1\}\times[0,1]\) (Region\({}_{1}\)) & SHADE \\  & \(p\) & 0.1 & fixed \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter scheme of PVCDE. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stage** & **Search Object** & **Parameter** & **Parameter Scope** & **Parameter Control Strategy** \\ \hline \multirow{4}{*}{stage1} & so\({}_{1}\) (DE/rand/1) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Regions) & jDE \\  & so\({}_{2}\) (DE/current-to-pbest/1) & \(F\) and \(CR\) & \(\{0.5,1\}\times[0.5,1]\) (Region\({}_{1}\)) & SHADE \\  & so\({}_{2}\) (DE/current-to-pbest/1) & \(p\) & 0.5 & fixed \\ \hline \multirow{4}{*}{stage2} & so\({}_{1}\)(DE/rand/1) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Regions) & jDE \\  & so\({}_{2}\) (DE/current-to-pbest/1) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Regions) & jDE \\ \cline{1-1} \cline{2-5}  & so\({}_{1}\)(DE/rand/1) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Regions) & jDE \\ \cline{1-1} \cline{2-5}  & so\({}_{2}\) (DE/current-to-pbest/1) & \(F\) and \(CR\) & \(\{0.1,1\}\times[0,1]\) (Region\({}_{1}\)) & SHADE \\  & so\({}_{2}\) (DE/current-to-pbest/1) & \(p\) & 0.1 & fixed \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameter scheme of PSCDE. The comparison can thus test whether the method of region division and combination is valid. The algorithms settings are as follows:  
* SHADE: The parameters for DE/current-to-pbest/1 operation are set to \(p\) = 0.1, archive-size = 2*_NP_, which is the same as that in the initial paper. The memory-size of the history list used by the parameter strategy is set as 6, referencing a previous study [24]. For parameter _NP_, we set the _NP_ as 4\(D\). * PVCDE: The population size is set to _NP_ = 5\(D\) for the purpose of grouping, with archive-size = 1.4*_NP_ and memory-size = 6. The initial group size ratios (Igsr) are set as [8/10, 1/10, 1/10], and the basic proportions (\(P_{base}\)) of all search objects in phase 1 are set to [7/10, 1/10, 1/10], which emphasizes so\({}_{1}\).