# Parameter Combination Framework for the Differential Evolution Algorithm
## 5 Customizing Two Parameter Combination Strategies
### Combine Parameter Regions or Values for Single-Operation DE  
This strategy we named PVCDE (Parameter Values Combination for DE) is designed for single-operation DE which means that the multiple search objects use the same evolutionary operation but with different parameter values (or regions). For the DE algorithm, we chose the mutation operation DE/current-to-pbest/1 proposed by the JADE algorithm, as follows:  
\[v_{i}=x_{i}+F\cdot(x_{pbest}-x_{i})+F\cdot(x_{r2}-x_{r1}) \tag{18}\]  
where \(x_{pbest}\) is an individual that is randomly selected from the top \(NP\times p\) (\(p\in[0,1]\)) sorted individuals in the current generation. \(x_{r2}\) is a random individual in the population, and \(x_{r1}\) is selected from the population and archive which contains the failed individuals from the selection step. The crossover strategy is binomial crossover, and the selection strategy is the same as the classic DE. The parameters include \(NP\), \(F\), \(CR\), \(p\), and _archive-size_. \(NP\) and _archive-size_ are set as fixed values, and \(F\), \(CR\), and \(p\) are controlled by region combination (for \(F\) and \(CR\)) and value combination (for \(p\)) method.  
In addition to the regions in Figure 3, we add two regions for \(F\) and \(CR\).  
* Region 5: \(F\)\(\in\)[0.1, 1], \(CR\)\(\in\)[0, 1]. This region consists of the entire range except for a very small \(F\).
* Region 6: \(F\)\(\in\)[0, 1], \(CR\)\(\in\)[0, 1]. This region consists of the entire range.  
We design three search objects. The first search object(so\({}_{1}\)) is used for wide range search with large \(F\), the second search object (so\({}_{2}\)) is used for local search with small \(F\) and \(CR\), and the third search object (so\({}_{3}\)) performs the most favorable search for current state by adaptive parameter values in the entire parameter domain. With the overlapping of the parameter regions, the search focus can be changed by adjust the group size of each search object. All the search objects adopt the SHADE parameter control strategy for \(F\) and \(CR\) within their own region. SHADE parameter control strategy is an adaptive strategy and its parameter distribution characteristic is analyzed in Section 3.  
We divide the evolutionary process into two stages. The first stage is an exploration stage, and the second stage is for exploitation. The design of search objects is showed in Table 2. In the initial stage, the search focus is exploration, thus we strengthen the region 1 by assigning larger group size to so\({}_{1}\), and weaken the region 3 and region 2 by assigning smaller group size to so\({}_{3}\) since its parameter distribution may move in the four regions. A so\({}_{2}\) with small group size achieves the purpose that region 4 cooperates with region 1. The grouping mode uses collaboration-based grouping method with predefined ratio of group size to \(NP\). After the initial stage, the search enters the exploitation stage, thus, we strength so\({}_{3}\) and weaken so\({}_{1}\) and so\({}_{2}\) by adaptive competition-based grouping. A slight difference between the two stages can be seen in the Table 2. so\({}_{1}\) changes from region 1 to Region 1\(\cup\)Region 2 in order to match a category of separable problem. so\({}_{3}\) varies from region 5 to region 6 to avoid minimal F value in the exploration stage. For parameter \(p\), 0.1 is a suggested value in relevant papers, we combine 0.1 with 0.5 by so\({}_{3}\) in exploration stage for strengthen the exploration for so\({}_{3}\).  
A little change for SHADE strategy also adopts in region 6. As in SHADE [20], the \(F_{i}\) for individual \(x_{i}\) is computed with Cauchy distribution as \(\text{randc}_{i}(M_{F,ri}\), 0.1). When \(\text{M}_{F,ri}\) is less than 0.05, we use \(1.1^{*}M_{F,ri}\) to replace 0.1. \(F_{i}\) is still restricted in the scope [0, 1] as SHADE does, but we have more chances to get a small value (in the interval from 0 to \(M_{F,ri}\)) for \(F_{i}\). The same method is used for \(CR\), but the minimum value of \(CR_{i}\) is limited as \(0.1/D\).