## 4 Tested Cases
### Numerical Examples

Table 3: The comparison of statistical optimization results. * Himmelbau function (HM) \[f(\mathbf{x})=\left({x_{1}}^{2}+x_{2}-11\right)^{2}+\left({x_{2}}^{2}+x_{1}-7\right)^ {2},x_{1,2}\in[-10,10]\] (31)
* Goldstein-Price function (GP) \[ \begin{array}{ll}f(\mathbf{x})=&[1+(x_{1}+x_{2}+1)^{2}(19-14x_{1}+3{x_{1}}^{2}- 14x_{2}+6x_{1}x_{2}+3{x_{2}}^{2})]\\ &\times[30+(2x_{1}-3x_{2})^{2}(18-32x_{1}+12{x_{1}}^{2}+48x_{2}-36x_{1}x2+27{x_ {2}}^{2})],x_{1,2}\in[-2,2]\end{array} \] (32)
* Generalized polynomial function (GF) \[ \begin{array}{ll}f(\mathbf{x})=u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\\ u_{i}=c_{i}-x_{1}(1-x_{2}^{i}),i=1,2,3\\ c_{1}=1.5,c_{2}=2.25,c_{3}=2.625,x_{1,2}\in[-5,5]\end{array} \] (33)
* Levy 3 function (L3) \[ \begin{array}{ll}f(\mathbf{x})=\sin^{2}(\pi\omega_{1})+\sum\limits_{i=1}^{2} (\omega_{i}-1)^{2}[1+10\sin^{2}(\pi\omega_{i}+1)]+(\omega_{3}-1)^{2}[1+\sin^{2 }(2\pi\omega_{3})]\\ \omega_{i}=1+\frac{x_{i}-1}{4},i=1,2,3,x_{i}\in[-10,10]\end{array} \] (34)
* Hartmann 3 function (H3) \[ \begin{array}{ll}f(\mathbf{x})=-\sum\limits_{i=1}^{4}\alpha_{i}\exp\left[-\sum \limits_{j=1}^{3}\beta_{ij}(x_{j}-p_{ij})^{2}\right]\\ 0\leq x_{1},x_{2},x_{3}\leq 1\end{array} \] (35)  
where  
\[\alpha=\left[ \begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array} \right]\beta=\left[ \begin{array}{cccc}3.0&10&30\\ 0.1&10&35\\ 3.0&10&30\\ 0.1&10&35\end{array} \right]P=\left[ \begin{array}{cccc}0.3689&0.1170&0.2673 \\ 0.4699&0.4387&0.7470\\ 0.1091&0.8732&0.5547\\ 0.0381&0.5743&0.8828\end{array} \right] \tag{36}\]
* Leon (LE) \[f(\mathbf{x})=100({x_{2}}-{x_{1}^{3}})^{2}+({x_{1}}-1)^{2};x_{1},x_{2}\in[-10,10]\] (37)  
The statistical results of 100 times for five AKBDO approaches are summarized in Table 3. In Table 3, the \(\mathrm{FE_{mean}}\) represents the mean of iteration times illustrating the efficiency of the method, while \(\mathrm{FE_{std}}\) denotes the variance of function evaluations, which can reflect the robustness of each method [58]. In Table 3, the numbers after the mean or standard deviation are the rank of the compared method for each numerical case. For example, 26.97/1 means the mean value is 26.97 while the method ranks first. The numbers marked in bold represent the first rank among the five AKBDO approaches. It can be inferred that the EW-LCB ranks first in most of the test problems, which indicates that the proposed EW-LCB outperforms the other compared approaches considering effectiveness. To further demonstrate the robustness of the proposed approach, Figure 5 plots the box plot of \(\mathrm{FE_{mean}}\) of all 100 runs. Figure 5: The box plot of the FEEan with different methods. Table 4 shows the average ranking of the performance of five AKBDO for all the tests. The average ranking of EW-LCB is the best among the five approaches. It is then followed with PLCB, LCB, EI, and WEI. When it comes to the robustness of the compared approaches, the proposed EW- LCB performs better than the PLCB, LCB, and WEI methods, while it is slightly inferior to the EI method. To evaluate whether the differences between the proposed EW-LCB method and the other four approaches are significant or not, the \(p\) values over multiple test cases are obtained by using the Bergmann-Hommel procedure [59]. The statistic test results are listed in Table 5. As shown in Table 5, all the \(p_{i}\)-values are less than 0.05, indicating that there are significant differences in the efficiency performance between the proposed EW-LCB and the other four approaches. To demonstrate the influences of initial sample sizes, the other two initial sample sizes were studied. The initial sample points are all generated by the LHD method, and function SA and L3 were selected as function SA needs a small sample size, while function L3 needs a large sample size. Table 6 shows the results of comparisons. The numbers after \({}^{\prime}\)/ represent the efficiency metric ranking of each method. It is easy to see that the initial sample sizes have a great influence on the function of SA.