## 5 Discussion  
From the comparative analysis in Section 4, it can be concluded that the proposed NSPSO provide solution that evenly and completely cover the solution of a RMOP, which is suitable for the candidate samples generation of an active learning reliability analysis. Significance test results in Tables 3, 4, 6 and 8 show that the NSPSO is statistically better than state-of-the-art algorithms. This is because the proposed NSPSO seized the characteristics of RMOPs that the solution is continuous. The normal search mode can lead the particles towards the potential solution region, avoiding the unwanted particle clustering. The dynamic particle repulsion strategy guarantees the evenness of the distribution of the particles. The particle memory strategy improves the convergency of the method. The density-based elite breeding strategy greatly reduce the computational cost. On the other hand, existing algorithms cannot treat RMOPs well because they are designed to deal with problems with finite optimal solutions. From the results of the two engineering case studies, another advantage of the proposed NSPSO can also be inferred, that the results of NSPSO  
Figure 13: Resulting optimal results for Engineer case 1: (**a**) NMMSO. (**b**) Multi_AMO. (**c**) DP-MSCC-ES. (**d**) RSCMSAESII. (**e**) NSPSO.  
is not influenced by the dimensions of the problem, which is important as problems with high dimensions are quite common in practice.  
Complexity of the normal search and dynamic repulsion strategy can be calculated as \(O(i\times d\times Np^{2})\), where \(i\) represent iteration times, d is the number of the variables. Computational complexity for particle memory strategy is \(O(i\times d\times Np)\). Complexity of density-based elite breeding strategy contains two parts, the selection process and breeding process, which can be calculated as \(O(j\times d\times Np)\) and \(O(j\times d\times Np\times B_{p}\times n_{b})\), respectively, in which \(j\) is the count of breeding iterations. In conclusion, the overall complexity of NSPSO is \(O(i\times d\times Np^{2}+i\times d\times Np+j\times d\times Np+j\times d\times Np \times B_{p}\times n_{b})\), and can finally be simplify to \(O(i\times d\times Np^{2})\), which is bigger than that of PSO (\(O(i\times d\times Np)\)). However, the density-based elite breeding strategy can greatly reduce the complexity of NSPSO by using much smaller initial population and increase the population size during the iteration.  
It should also be noted that, we chose PSO with niching technique as the base of our algorithm because of its concise particle behavior pattern and stable performance. But the framework of the proposed NSPSO especially the normal search pattern is universal, which can be integrated with various meta-heuristic algorithms, for example, Butterfly Optimization Algorithm (BOA) [39], Trees Social relations optimization algorithm (TSR) [40], and Whale optimization algorithm (WOA) [41].  
There are also two limitations of the proposed NSPSO in dealing with candidate samples generating problems in active learning reliability analysis: first, for problems with highly nonlinear LSF, the population size of NSPSO should be big enough to cover the LSF, as if the population size is small, details of the LSF may be lost; and second, breeding strategy and repulsion adjustment procedure should be further improved in order to deal with high dimensions and highly complex problems.