## 3 The Proposed Method
### (2)  
LDL strategy  
In the process of population evolution, the effective local exploitation of subswarms can better improve the overall performance of the algorithm. However, making full use of the local optimal information can speed up the development of the local areas. Using the idea of dimension learning for reference, this paper constructed an update template through the historical optimal dimension information of the particles and the global optimal dimension information of the population to update the particle speed and position, as shown in Equations (11) and (12):  
\[v_{i,j}^{(t+1)}=wv_{i,j}^{(t)}+c_{1}r_{1,j}(LDPbest_{i,j}^{(t)}-x_{i,j}^{(t)}) \tag{11}\]  
\[LDPbest_{i,j}^{(t)}=\left\{ \begin{array}{c}\mathit{Pbest}_{i,j}\quad(rand>Pl _{i})\\ \mathit{Lbest}_{k,j}\quad(rand\leq Pl_{i})\end{array} \right. \tag{12}\]  
The selection probability \(Pl\) is used to manage the probability that the dimension information of the \(x_{i}\) historical optimal position \(\mathit{Pbest}_{i}\) and its subswarm historical optimal position \(\mathit{Lbest}_{k}\) are selected. In this paper, when updating the particle speed, we put \(\mathit{Pbest}_{i}\) and \(\mathit{Lbest}_{k}\) in the same important position, so we assigned \(Pl\) = 0.5. In detail, for each dimension of \(x_{i}\), if the generated random number \(rand\) is greater than the selection probability \(Pl\), the information of the corresponding dimension in \(\mathit{LDPbest}_{i}\) is selected from \(\mathit{Pbest}_{i}\); otherwise, the \(\mathit{Lbest}_{k}\) where particle \(x_{i}\) belongs is selected. The MCL strategy can extract more individual information, maintain better diversity of the population, and promote particle exploration but it also limits the exploitation ability. The LDL strategy strengthens the particle exploitation ability through the information of the particle's _pbest_ and _Lbest_ of the subpopulation where the particle belongs, however, it is also easier to fall into a local optimization. To balance the advantages and disadvantages of the MCL strategy and LDL strategy, the UL strategy was proposed. Randomly select \(H(H\leq sn)\) subswarms, and use the _Lbest_ of the selected subswarms to construct a unit learning example _Ubest_, that is, the center of the historical optimal position of all the selected subswarms is used to guide the evolution of the target particles. The particle's speed update is affected by the combination of the particle's _pbest_ and _Ubest_, the best location in the history of the individual, and the update formulas are Equations (13) and (14):  
\[v_{i,j}^{(t+1)}=wv_{i,j}^{(t)}+c_{1}r_{1,j}(\textit{Pbest}_{i,j}^{(t)}-x_{i,j} ^{(t)})+c_{2}r_{2,j}(\textit{Ubest}_{j}^{(t)}-x_{i,j}^{(t)}) \tag{13}\]  
\[\textit{Ubest}=\frac{1}{H}\sum_{h=1}^{H}\textit{Lbest}_{h} \tag{14}\]  
At different evolutionary stages, learning examples have different effects on the population evolution. For example, at the initial stage of the evolution, selecting the MCL learning model can meet the needs of the algorithm to fully explore the search space at the initial stage. At the end of evolution, the LDL strategy was adopted to accelerate the convergence of the algorithm. Therefore, to meet the needs of different evolutionary stages of the algorithm and give better play to the advantages of the multiple learning templates, this paper proposed a learning model selection mechanism with reward and punishment mechanisms. By selecting the benchmark function and the reward and punishment factors, the benchmark function ensures the overall selection probability change trend of the different templates in the evolution, and the reward and punishment factors fine tune the selected probability of the templates according to the evolutionary stage. There is more effective control of the probability that different learning templates are selected in the iteration to affect the evolution process of the population to improve the performance of the algorithm.