# Personalized-Template-Guided Intelligent Evolutionary Algorithm
## 2 Methods
### Template Generation Stage
#### 2.1.3 Personalized Template Generation Strategy Based on Particle Dimensional Distribution  
There are three kinds of dimensional distribution of a particle: the first is concentrated distribution, the second is similar to normal distribution, and the third is random distribution. In the first two cases, just finding the mean point of the dimensional distribution can help the particle converge quickly. Therefore, we put forward the particle dimensional concentration degree to judge the particle dimensional distribution. For the particle that meets one of the first two conditions, all dimension values of the personalized template are the same, so as to guide the particle to mine near this dimension value, and then get the mean point of particle dimension distribution through the elite selection strategy. We use the parameter \(\delta\) to regulate the value of \(fd_{i}\). At the initial stage of optimization, the particle dimension distribution is not stable, so \(fd_{i}\) should be small. With the progress of iteration, under the elite selection strategy, the dimension distribution of particles will gradually stabilize. At this time, the personalized guidance template can be made for the particle according to its dimensional distribution to help mine their potential knowledge. At the later stage, the dimension distribution of the particle itself has been particularly stable. At this time, each dimension should be optimized and converged separately, so the value of \(fd_{i}\) should be reduced. \[fd_{i}= \begin{cases}\delta,&\text{if $w_{i}$= $0$}\\ (1-\frac{s_{i}}{w_{i}})\times\delta,&\text{otherwise}\end{cases}  \tag{8}\]  
\[\delta=\sin\biggl{(}\frac{t\times\pi}{T}\biggr{)} \tag{9}\]  
\[s_{i}=abs(\frac{1}{D}\times\sum_{j=1}^{D}x_{i}^{j}-\frac{2}{D}\times\sum_{j=1 }^{D/2}x_{ir}^{j}) \tag{10}\]  
\[w_{i}=abs(\max(x_{i})-\min(x_{i})) \tag{11}\]  
where \(fd_{i}\) is the particle dimension concentration, \(x_{i}\) is the \(i\)-th particle, \(\delta\) is the concentration control parameter, and \(x_{ir}\) is a new particle obtained by disrupting the dimension order in \(x_{i}\). When \(fd_{i}\) is large, it is more likely to generate templates with the same dimension values. In addition, the dimension value the template chooses for knowledge guidance is the selection of the \(i\)-th particle of the parent in the corresponding template block with a certain probability. If the historical particle selection value is empty, it will be generated randomly. See Equations (12) and (13) below for the specific dimension value selection. \[Mi^{L}= \begin{cases}1,&\text{if $r<fd_{i}$ or }\ (\beta=1\text{ and }r<(\frac{t}{T})^{2}\\ \\ 2,&\text{if $r>=fd_{i}$ or }\ (\beta=1\text{ and }r<(\frac{t}{T})^{2}\\ \\ &\text{and $PM_{i}^{L}=2$})\end{cases}  \tag{12}\]  
where \(Mi^{L}\) represents the dimension learning mode selected when the \(i\)-th particle generates the template \(L\). If \(Mi^{L}=1\), a template with the same dimension value is generated. We set judgment condition 2 in our expression to be superior to condition 1. Where \(r\) is a random number between 0 and 1. \(PM_{i}^{L}\) represents the mode of dimension learning selected when the \(i\)-th parent particle generates the template \(L\). \(L\) is a random integer of 1 or 2. The history-retrace template is selected when \(L=1\), and the value-driven template is selected when \(L=2\). \[rS_{i}^{L}= \begin{cases}prS_{i}^{L},&\text{if $\beta=1$ and $r<(\frac{t}{T})^{2}$ and $\exists prS_{i}^{L}\in\mathbb{R}$}\\ r8,&\text{otherwise}\end{cases}  \tag{13}\]  
where \(rS_{i}^{L}\) represents the dimension selected by the \(i\)-th particle when generating the \(L\)-type template. \(prS_{i}^{L}\) represents the dimension selected by the \(i\)-th parent particle when generating the \(L\)-type template. \(r8\) is a random integer from 1 to \(D\). 1 Generate a history-retrace template  
\[a_{i}^{j}= \begin{cases}f1_{i}^{r5}+(f2_{i}^{r5}-f1_{i}^{r5})\times r,&\text{ if $M_{i}^{1}=1$}\\ f1_{i}^{j}+(f2_{i}^{j}-f1_{i}^{j})\times r,&\text{if $M_{i}^{1}=2$}\end{cases}