## 3 Experimental Results and Discussion
### Comparative Analysis of the Optimizing Diversity of the Parallel EGO Algorithm  
#### 3.3.1 Population Diversity  
When an optimization algorithm is prone to getting trapped in local optima, strategies can be employed to provide population diversity, helping the algorithm escape local optima and thereby increasing the probability of finding higher-quality solutions [33]. The classical EGO algorithm inherently achieves a perfect balance between local exploitation and global exploration; however, it is also prone to getting trapped in local optima. The introduction of the parallel EGO algorithm not only enhances the efficiency of the entire optimization process but also, by multiplying the corresponding number of IFs after the original EI function, can be seen as increasing the diversity of the optimization. Therefore, the impact of different point-filling quantities on the performance of the parallel EGO algorithm is analyzed from the perspective of algorithmic diversity. The calculation method for population diversity is as follows:  
\[Pop_{div}(l)=\frac{1}{n}{\sum_{i=1}^{n}\sqrt{\sum_{j=1}^{D}\Bigl{(}\overline{ pos_{j}}-pos_{i,j}(l)\Bigr{)}^{\frac{1}{2}}}},\overline{pos_{j}}=\frac{1}{n}{ \sum_{i=1}^{n}pos_{i}(l)} \tag{24}\]  
Figure 11: The process for searching multi-points using the PEI criterion within a single optimization cycle. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Systems**} & **Test** & \multirow{2}{*}{**Stopping Criterions**} & **Number** & \multirow{2}{*}{**2 Points**} & \multirow{2}{*}{**4 Points**} & \multirow{2}{*}{**6 Points**} & \multirow{2}{*}{**8 Points**} & \multirow{2}{*}{**10 Points**} \\ \cline{5-8}  & & & & Median & 59.00 & 28.50  
where \(n\) and \(D\) represent the total population size and the number of design variables, respectively. \(pos_{i,j}(l)\) represents the position of the \(i\) individual in the \(j\) dimension, and \(\overline{pos}_{j}\) denotes the average position of all individuals in the \(j\) dimension. A low population diversity indicates that the search is exploring the design space near the local optimum. In contrast, a high population diversity suggests that the search is exploring regions of the design space farther from the local optimum. Figures 12 and 13 present the average population diversity curves of the parallel EGO algorithm based on the OK model and the GEK model, respectively, under different point-filling quantities during optimization process. For the case based on OK model, the parallel EGO algorithm with a larger point-filling quantity generally exhibits greater population diversity in the early stages of optimization, indicating that it explores a larger design space. As the optimization process progresses, population diversity gradually decreases. Meanwhile, the rate of decline in population diversity is relatively slower for the parallel EGO algorithm with a larger point-filling quantity. This also explains why its overall solution convergence speed is relatively slower (Figure 8). However, the advantage of having greater population diversity is that it can facilitate the parallel EGO algorithm in escaping local optima, thereby increasing the probability of discovering higher-quality optimal solutions. For example, in the case of test functions F2, F3, F5, and F7, higher-quality solutions were found when using a larger number of filled points. For the case based on the GEK model, overall, the impact of different point-filling numbers on the population diversity of the parallel EGO algorithm is reduced, except for test functions F1, F3, and F8. Combining the average convergence curves for the corresponding test functions in Figure 6, it can also be observed that the differences in the impact of different point-filling numbers on the convergence speed of the parallel EGO  
Figure 12: Population diversity curves of the OK-based parallel EGO algorithm on all tested problems. algorithm are minimal.