## 3 Experimental Results and Discussion
### Comparative Analysis of the Optimization Performance of the Parallel EGO Algorithm
#### 3.1.2 Convergence Speed of Optimization  
To investigate the impact of different point-filling quantities on the convergence speed of the parallel EGO algorithm, Figures 8 and 9 present the average convergence curves of the parallel EGO algorithm based on the OK and GEK models, respectively, under different point-filling strategies. From Figure 8, it can be observed that for the case based on the OK model, the convergence speed varies significantly across different point-filling quantities for almost all test functions, except for F3. A point-filling quantity of two exhibits an absolute performance advantage in terms of convergence speed for test functions F1, F4, F6, F8, and F9. This conclusion can be drawn from the fact that the average convergence curve for a point-filling quantity of two remains consistently below the other average convergence curves throughout the optimization process. In contrast, the average convergence speed for a point-filling quantity of 10 is relatively slower across most test functions. The average convergence speed for other point-filling quantities exhibits a certain degree of fluctuation between these two cases. This result is generally consistent with the findings on the accuracy of the optimized solutions presented above. This indicates that the optimization performance of the parallel EGO algorithm based on OK model is most reliable when the point-filling quantity is set to two. However, as the point-filling quantity gradually increases to 10, the stability of the parallel EGO algorithm's optimization performance deteriorates significantly. From the average convergence curves in Figure 9, it can be observed that in the case of the GEK model, the impact of different point-filling quantities on the convergence speed of the parallel EGO algorithm differs only in the early stages of the optimization process for almost all test functions, except for F3 and F8. Moreover, during this stage, the impact on the average convergence speed exhibits a trend that is essentially linearly correlated with  
Figure 7: Three-dimensional maps for two-dimensional Rastrigin and rotated Rastrigin functions. the increase in point-filling quantity. As the optimization process progresses, the differences in average convergence speed among different point-filling quantities gradually diminish. The potential reason for this phenomenon is that, in the early stages of the optimization process, fewer point-filling quantities lead to more frequent updates of the surrogate model. This provides more reliable design space information for the EGO algorithm's optimization. As the optimization process progresses and considering that GEK model has superior fitting performance compared to OK model, it can rapidly compensate for the loss of design space information due to fewer surrogate model updates in the early stages of the optimization process. This result also indicates that the convergence performance of the parallel EGO algorithm based on the GEK model is more stable and reliable. It is noteworthy that both OK-based and GEK-based parallel EGO algorithms demonstrate relatively poor convergence performance on test function F8 (rotated expanded Griewank plus Rosenbrock function), exhibiting slower convergence speeds. Analysis of F8's functional characteristics reveals its highly oscillatory and multimodal nature, which can lead to significant deviations between the PEI function and the true EI function. Consequently, the PEI criterion may prove particularly susceptible to failure when applied to functions of this category. As evidenced in Figures 8 and 9, the GEK-based parallel EGO algorithm demonstrates superior convergence performance over its OK-based counterpart on test functions F1, F4-F7, and F9. This enhancement stems from GEK's effective utilization of gradient information from the same number of sample points, enabling more comprehensive exploration of the design space when identifying promising candidate points.