## 3 Experimental Results and Discussion  
### Comparative Analysis of the Optimization Performance of the Parallel EGO Algorithm  
#### 3.1.1 Optimization Solution Accuracy  
A comparative analysis was conducted to examine the impact of different point-filling quantities on the optimization solution accuracy of the parallel EGO algorithm based on the OK model and the GEK model within a limited total number of evaluations. Table 2 presents the statistical results of the optimized solutions for all test cases, along with the Friedman ranking across the test cases [32]. A \(p\)-value less than 0.05 indicates statistically significant differences in solution accuracy between different point-filling strategies at the specified significance level, whereas values exceeding this threshold demonstrate no statistically significant performance variations. The results indicate that, for the case based on the OK model, filling two update points per optimization cycle achieved the most accurate converged solutions for test functions F1, F4, F6, F8, and F9. Furthermore, the significance levels (\(p\)-values) reveal statistically significant differences in solution accuracy among different point-filling strategies for these test functions. Additionally, for F5, it also obtained the second-highest ranked approximate optimal solution in terms of accuracy. The observed phenomenon may be attributed to the high-dimensional, multimodal characteristics of these functions, where a relatively small number of filling points effectively avoids local optima traps while maintaining convergence efficiency by preventing excessive exploration. In contrast, as the number of filled points increased to 10, the algorithm yielded the least accurate converged solutions for almost all test functions, except for F3 and F6. This suggests that the stability of the optimization performance of the parallel EGO algorithm based on the OK model deteriorates as the number of filled points increases. However, overall, as the number of filled points increased from 2 to 10, the change in the accuracy of the converged solutions did not follow a strictly linear decline; instead, there were fluctuations within certain intervals. For test functions F2 and F6, the performance with six filling points proves superior to that with four points. This phenomenon similarly manifests in functions F5 and F9, where eight filling points demonstrate better performance than six. The underlying rationale may lie in the fact that intermediate quantities (6-8 points) in certain specific problems enable the parallel EGO algorithm to escape particular types of local optima traps while maintaining an optimal balance between global exploration and local exploitation. It is noteworthy that the impact of point-filling quantity on parallel EGO algorithm performance varies significantly across different functions. For instance, the performance differences between various filling quantities nearly disappear when comparing the original [PERSON]'s function (F2) with its rotated version (F3). This phenomenon likely stems from how rotation weakens the original function's variable coupling structure (as illustrated in Figure 6, showing 3D maps of both versions), thereby reducing optimization process sensitivity to sampling strategies. A similar pattern emerges between the original [PERSON]'s function (F6) and its rotated counterpart (F7). In contrast, for Rastrigin functions (F4, F5), the algorithm's performance remains highly sensitive to filling quantities regardless of rotation, with two-point filling demonstrating consistent performance--a characteristic potentially related to the function's strong multimodality (visualized in Figure 7's 3D maps). An appropriate filling quantity balances avoidance of local optima traps with maintained exploration efficiency.