## 4 Numerical Examples
### Explicit Performance Functions  
Four explicit performance functions were used for comparative study. Two of them are two-dimensional and the other two are three-dimensional. Corresponding equations and state variable ranges are shown in Table 2.  
Since LSF is the region where performance function equals zero, in order to transform the LSF-finding problem to a RMOP, the first step here is to take the absolute value of the performance function in Table 2 while keeping other settings fixed. Then the proposed NSPSO and the four competitive algorithms were employed to solve the transformed RMOPs. Resulting solution were visualized in Figure 7, in which the red dots are the found optimal points while the lines and surfaces are the true optimal regions (i.e., the real LSF). The figures show the converged result of each algorithm optimization for each test function. It can be seen that the optimal solution obtained by NSPSO can cover the entire LSF evenly. With more found optimal points and better distribution uniformity, there is a bigger chance that more sampling points were found near the LSF, which may indicate a more accurate LSF predicted by the surrogate model. On the other hand, due to the clustering trend and ignoring the continuity of the solution of RMOP, all competitive algorithms have very poor performance. The reason for these results is that the proposed NSPSO take the continuity of the solution of RMOP into consideration and employ a multi-strategy framework to ensure the uniformity and diversity of the solution. Moreover, higher dimensions make the competitive algorithms perform more poorly, since an extra dimension brings more spatial complexity for RMOPs, whereas the performance of NSPSO was almost unaffected. With a better training sample, the resulting surrogate model can certainly have a better approximation of the LSF. \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Function Equation** & \(\mathbf{d}\) & **State Variable Range** \\ \hline \(1\) & \(Z_{1}\)\(=\)\(G_{1}(x_{1},x_{2})=\min\left\{\begin{array}{c}3+0.1(x_{1}-x_{2})^{2}-\frac{(x_{1}+x_{2})}{ \sqrt{2}};\\ 3+0.1(x_{1}-x_{2})^{2}+\frac{(x_{1}+x_{2})}{\sqrt{2}};\\ (x_{1}-x_{2})+\frac{6}{\sqrt{2}};\\ (x_{2}-x_{1})+\frac{6}{\sqrt{2}};\end{array}\right.\) & \(2\) & \(x_{1}\in[-6,\ 6]\) \\ \(2\) & \(Z_{2}\)\(=\)\(G_{2}\left(x_{1},x_{2}\right)=1-(4-2.1x_{1}^{2}+x_{1}^{4}/3)\times x_{1}^{2}\) & \(2\) & \(x_{1}\in[-2,\ 2]\) \\ \(+x_{1}\times x_{2}+(4x_{2}^{2}-4)\times x_{2}^{2}\) & \(3\) & \(x_{2}\in[-1.2,\ 1.2]\) \\ \(3\) & \(Z_{3}\)\(=\)\(G_{3}\left(x_{1},x_{2},x_{3}\right)=\left(\cos\left(x_{2}\right)^{2}-e^{x_{1}^{ 2}-10}-\sin\left(x_{1}+x_{2}\right)-x_{3}\right)^{2}\) & \(3\) & \(x_{1}\in[-2,\ 2]\) \\ \(4\) & \(Z_{4}\)\(=\)\(G_{4}\left(x_{1},x_{2},x_{3}\right)=\left(100-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\right)^{2}\) & \(3\) & \(x_{1}\in[-10,\ 10]\) \\ \(x_{2}\in[-10,\ 10]\) & & \(x_{2}\in[-10,\ 10]\) \\ \(x_{3}\in[-10,\ 10]\) & & \(x_{3}\in[-10,\ 10]\) \\ \hline \end{tabular}
\end{table} 
Table 2: Basic information about the test functions. To have a further quantitative comparison of NSPSO and competitive algorithms, _IGD_S_ and _EOR_ were calculated for all resulting data. The Wilcoxon ranksum [35] test was conducted, in which the significance level was set to 0.05. Calculation results are shown in Tables 3 and 4. We use Symbols \"+\", \"\(-\)\", and \"\(-\)\" to represent that NSPSO is significantly better than, worse than, or similar to competing algorithms, respectively. The best algorithm for each problem was highlighted in the tables.