## 3 Methodology
### Clone  
As shown in Algorithm 1, the clone steps aim to enhance the diversity and convergence of the external archive \(A\), thereby accelerating the PSO convergence speed, particularly for the parameter-independent MAPSO algorithm. The process involves the following steps: initialization, SDE distance calculation, selection and duplication, and generation of offspring population. The clone process is illustrated in Figure 2. Below are the detailed descriptions.  
Figure 1: Illustration of the MAPSO process for solving data transmission tasks.  
Initialization: First, copy the external archive \(A\) to a temporary archive \(B\) and initialize a temporary archive \(C\) as empty. Make sure that Archive \(A\) remains intact in the following steps.  
Shift-based density estimation (SDE) distance calculation: SDE is an improved density estimation strategy designed to address the challenges faced by Pareto-based evolutionary multi-objective optimization algorithms when handling multi-objective optimization problems. Traditional density estimation methods only reflect the distribution of individuals in the population, while SDE considers individuals' distribution and convergence information. Calculate the SDE distance for each solution \(b_{i}\) in \(B\). The SDE distance measures the diversity of the solutions. Refer to the relevant literature for specific calculation steps and equations [22].  
```
Input: External repository Output: New temporary solution set \(S\)
1. \(B\gets A\)
2. \(C\leftarrow\varnothing\)
3. AssignSDE\((B)\)
4. sort\((B)\) // Sort \(B\) based on the SDE metric
5. ifLength\((B)\)\(>NC\)then
6. \(B=B[:,\ NC]\)
7. endif
8. for\(i=1\) to Length\((B)\)do
9. Calculate \(n_{i}\)
10. Create \(n_{i}\) solutions for \(b_{i}\) and append to \(C\)
11. endfor
12. S\(\leftarrow\)modify(C)
13. Return \(S\)
```  
**Algorithm 1** Clone Process  
Figure 2: Illustration of a clone.  
Selection and duplication: Next, select \(NC\) solutions from the temporary archive \(B\) for duplication. The number of duplications \(n_{i}\) for each solution \(b_{i}\) is calculated using the following (14):  
\[n_{i}=\left\lceil N\times\frac{SDE(b_{i})}{\sum_{i=1}^{\text{NC}}SDE(b_{i})}\right\rceil \tag{14}\]  
where \(SDE(b_{i})\) presents the \(SDE\) distance of the solution \(b_{i}\), and \(N\) is the total number of solutions to be duplicated. This step ensures that solutions with more considerable SDE distances obtain more duplication opportunities, thereby enhancing the diversity and convergence of the population.  
Generation of offspring population: Each solution \(b_{i}\) is duplicated \(n_{i}\) times and added to the temporary archive \(C\), forming a new offspring population \(C\). This population then undergoes evolutionary search, applying operations such as crossover and mutation, resulting in a new offspring population \(S\). The clone selects superior solutions based on SDE distance, ensuring that the external archive \(A\) maintains good diversity and convergence, effectively guiding the evolution of population \(P\) in PSO and avoiding a local optimum. The evolutionary search strategy in MAPSO is implemented in the same manner as in NSGA-II [23].  
The clone guarantees the population's diversity and convergence and accelerates MAPSO convergence speed. Through these steps, the external archive \(A\) exhibits excellent diversity and convergence, promoting the practical evolution of the MAPSO population \(P\), making the algorithm more efficient in handling multi-objective optimization problems.