## 3 Novel Approach to Computing Infill Criteria for MOEGO
### A Brief Description of the Monte Carlo Approach
In order to investigate the speedup effect of the number of Monte Carlo sample points, the GMOEGO algorithm with different amounts of Monte Carlo sample points, \(1\times 10^{2}\), \(1\times 10^{3}\), \(1\times 10^{4}\), \(1\times 10^{5}\), and \(1\times 10^{6}\), is validated, and the traditional MOEGO algorithm is taken as the reference optimizer for comparison. For the purpose of saving testing time, one test function, the ZDT1 function with 2 variables (\(d=2\)), is selected for analyzing the performance of GPU implementation. The test results for the speedup ratio of ZDT1 are listed in Table 1. It can be learned that the most time-consuming part, up to 99 percent of the overall costs (the MOEGO algorithm with \(1\times 10^{6}\) Monte Carlo sample points), are really spent in MCHVI calculations (see the fourth column of Table 1), the time consuming of GMOEGO presented are now speedup by proposed GPU implementation, and significant acceleration, up to 12.575 speedup ratio, are achieved (see the last column in Table 1). In order to make it clear, the corresponding speedup ratio of the total cost to a different number of Monte Carlo sample points is illustrated in Figure 6. When the Monte Carlo sample points are few, it fails to have GPU speedup (see Figure 6) due to the fact that the cost of data exchange between the CPU and GPU side is mainly dominated in such circumstances, as listed in Table 1. It can also be observed that a continual increase in speedups related to the growing number of sample points can be achieved, which are favorable for coping with the problems in the case of a large number of sample points required. Such situations usually occur in complex MOO problems under the requirement of high accuracy in capturing the Pareto front. Figure 6: Speedup ratio of total cost to different numbers of Monte Carlo sample points. ### Numerical Tests of the GMOEGO Algorithm

In order to investigate the performance of the proposed GMOEGO algorithm, all six test functions (see Appendix A) are selected as the test functions. In order to mimic real engineering optimizations, test functions with more variables are constructed: 10 variables for ZDT1-ZDT3 functions, 5 variables for DTLZ2 function, 8 variables for DTLZ5 function, and 10 variables for DTLZ7. In contrast, these test cases are also optimized by the referenced MOEGO algorithm, which is run on a 12-core CPU (Intel Core i9-9900k) in parallel. The Monte Carlo sample points of the GMOEGO algorithm are set as \(n_{p}=1\times 10^{5}\) for two-objective functions (ZDT1-ZDT3), and \(n_{p}=1\times 10^{6}\) for three-objective functions (DTLZ2, DTLZ 5, and DTLZ7). Both algorithms are performed on the same computational platform mentioned at the beginning of this section. The test results are shown in Table 2. The Pareto solution distributions compared with the theoretical true Pareto front are illustrated in Figure 7. It can be noticed that, for all the test cases, the Pareto solution distributions obtained by the GMOEGO algorithm and the referenced MOEGO algorithm are very close to the theoretical true Pareto front, and the obtained IGD values listed in the third column of Table 2 are around \(10^{-3}\) to \(10^{-5}\), which indicates the optimized Pareto fronts are very close to the theoretical Pareto fronts. The total wall-clock time costs and the MCHVI calculation costs are given in the fourth and fifth columns of Table 2, and their corresponding comparison histories are illustrated in Figure 8. It can be seen that, as analyzed in Section 3, the computational cost of the MCHVI calculations is the main part of the overall optimization procedure, and it is not hard to imagine that improving this module of the optimization algorithm will bring the most benefits, and as expected, the computational cost improvements of the proposed GMOEGO algorithm mainly come from the speedup of the MCHVI calculations.