# A Modified Cloud Particles Differential Evolution Algorithm for Real-Parameter Optimization
## 3 Modified Cloud Particles Differential Evolution Algorithm
### Control Parameters Assignments  
In classic DE, control parameters are preset and fixed during the entire iteration process. However, it is impossible to find one constant parameter setting that can fit all problems. As pointed out in [53], the different parameter settings not only play an important role in the performance of DE, but also may be used to solve specific test problems. Thus, a novel parameter adaptation scheme is presented to adjust the parameter \(F\) and \(Cr\) at different evolutionary stage. In MCPDE algorithm, the parameter settings are divided into three stages according to the successful mutation factors \(F\) at current generation. The initial \(F_{i}\) and \(Cr_{i}\) used by each cloud particle \(\mathbf{x}_{i}\) are generated independently and formulated as follows, respectively:  
\[F_{i}=r_{2}\times(r_{1}\times\frac{f_{0}}{5\sqrt{MaxFES}}+\frac{f_{0}}{5})+f_{0} \tag{9}\]  
Figure 1: Variation curves of inertia factors. (**a**) the variation of \(\omega_{1}\); (**b**) the variation of \(\omega_{2}\). \[Cr_{i}=r_{2}\times(r_{1}\times\frac{cr_{0}}{5\sqrt{MaxFES}}+\frac{cr_{0}}{5})+cr_{0} \tag{10}\]  
where \(f_{0}\) and \(cr_{0}\) are initialized to be 0.5, respectively. \(r_{1}\) and \(r_{2}\) are random numbers in [0, 1]. In each generation, the set \(S_{F}\) is used to store all successful mutation factors at current generation. Similarly, the set \(S_{Cr}\) stores all the successful crossover rates at current generation. The size of \(S_{F}\) is recorded as \(\left\lvert\,S_{F}\,\right\rvert\). If \(\left\lvert\,S_{F}\,\right\rvert\) exceeds the current population size \(N\), randomly selected elements are deleted from \(S_{F}\) and \(S_{Cr}\). Then \(S_{F}\) and \(S_{Cr}\) are preserved for the next generation. When the set \(S_{F}\) is empty, it indicates that \(F\) and \(Cr\) at current generation are the proper parameters for the algorithm. Then, they are preserved for the next generation. When \(\left\lvert\,S_{F}\,\right\rvert\) is less than the current population size \(N\), new control parameters \(F^{\prime}_{1},F^{\prime}_{2},\cdots,F^{\prime}_{N-\left\lvert\,S_{F}\,\right\rvert}\) are produced according to Equation (9). \(Cr^{\prime}_{1},Cr^{\prime}_{2},\cdots,Cr^{\prime}_{N-\left\lvert\,S_{F}\,\right\rvert}\) are defined by  
\[Cr^{\prime}_{i}=\sigma\left(S_{Cr}\right)+r_{i}\ (i=1,\cdots,N-\left\lvert\,S_{Cr}\, \right\rvert) \tag{11}\]  
where \(\sigma\left(S_{Cr}\right)\) refers to the standard deviation of \(S_{Cr}\). \(r_{i}\) is randomly selected in the range [0, 1]. By the end of each generation, the parameters \(F\) and \(Cr\) are updated when \(\left\lvert\,S_{F}\,\right\rvert\) is less than the current population size \(N\), as defined by  
\[F=S_{F}\cup F^{\prime} \tag{12}\]  
\[Cr=S_{Cr}\cup Cr^{\prime} \tag{13}\]  
where \(F^{\prime}=(F^{\prime}_{1},F^{\prime}_{2},\cdots,F^{\prime}_{N-\left\lvert\,S _{F}\,\right\rvert})\), \(Cr^{\prime}=(Cr^{\prime}_{1},Cr^{\prime}_{2},\cdots,Cr^{\prime}_{N-\left\lvert \,S_{F}\,\right\rvert})\). In MCPDE algorithm, different control parameters are chosen at different stages. At the early stage of evolution, the control parameter values near \(f_{0}\) and \(cr_{0}\) with the randomization according to Equations (9) and (10). Better diversity may improve the exploration ability. Then, cloud particles try to explore further areas in the solution space. At each generation, better control parameters are preserved for the next generation. The population diversity is improved and the convergence speed is accelerated with better control parameters. However, it is difficult to find better control parameters with the increasing generation. Thus, the algorithm may hard to jump out of the local optimum because of faster convergence and poorer diversity. In order to solve these problems, some new parameters \(F\) are introduced to maintain search efficiency according to Equations (9) and (12) while some new parameters \(Cr\) are produced to improve population diversity according to Equations (11) and (13).