## 4 Tested Cases  
### Numerical Examples  
In this subsection, ten widely used benchmark problems from Ref. [33; 50; 51; 52] are used to illustrate the effectiveness of the proposed EW-LCB method. Meanwhile, four famous AKBDO approaches, EI, WEI, LCB, and PLCB, are tested to compare with the EW-LCB method. As the optimal solutions for all benchmark problems can be obtained, the terminal condition is defined such that the relative \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **EI** & **WEI** & **LCB** & **PLCB** & **EW-LCB** \\ \hline Mean Value & 7.64 & 7.86 & 7.98 & 7.66 & **6.48** \\ Standard deviations & 1.352 & 1.498 & 1.301 & 1.780 & **0.505** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison results of different approaches of the illustrated example. Figure 4: The final result of the proposed approach on the illustrated function. error between the optimal solution obtained from the Kriging model and the true one is within 0.002. Therefore, the number of iterations is regarded as the merit of effectiveness. Considering the randomness of the results, many AKBDO approaches repeat their algorithms dozens of times and provide statistical results [53; 54]. Furthermore, some approaches use the deterministic sampling and optimization algorithms such as Hammersley and deterministic PSO [55; 56; 57] to avoid the randomness. In this work, each method ran 100 times with different initial samples and their statistical results are recorded in Table 3.  
The expressions of benchmark problems are listed as,  
* Peaks function (PK) \[f(\mathbf{x})=3(1-x_{1})^{2}e^{-x_{1}2-(x_{2}+1)^{2}}-10(\frac{x_{1}}{5}-{x_{1}}^{ 3}-{x_{2}}^{5})e^{-x_{1}2-x_{2}2}-\frac{1}{3}e^{-x_{2}2-(x_{1}+1)^{2}},x_{1,2} \in[-3,3]\] (27)
* Banana function (BA) \[f(\mathbf{x})=100{({x_{1}}^{2}-{x_{2}})}^{2}+(1-x_{1})^{2},x_{1,2}\in[-2,2]\] (28)
* Sasena function (SA) \[f(\mathbf{x})=2+0.01(x_{2}-{x_{1}}^{2})^{2}+(1-x_{1})^{2}+2(2-x_{2})^{2}\ +7 sin(0.5x_{1})sin(0.7x_{1}x_{2})\,x_{1,2}\in[0,5]\] (29)
* Six-hump camp-back function (SC) \[f(\mathbf{x})=(4-2.1{x_{1}}^{2}+\frac{x_{1}^{4}}{3}){x_{1}}^{2}+x_{1}x_{2}+(-4+4{ x_{2}}^{2}){x_{2}}^{2},x_{1,2}\in[-2,2]\] (30) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Functions** & **Items** & **EI** & **WEI** & **LCB** & **PLCB** & **EW-LCB** \\ \hline \multirow{3}{*}{PK} & FE\({}_{\text{mean}}\) & 29.82/3 & 30.13/4 & 29.68/2 & 31.99/5 & **26.97/1** \\  & FE\({}_{\text{std}}\) & 2.435/2 & 2.884/3 & **2.068/1** & 5.107/4 & 5.34/5 \\  & FE\({}_{\text{mean}}\) & 33.23/3 & 32.15/2 & 33.88/4 & 34.34/5 & **26.33/1** \\  & FE\({}_{\text{std}}\) & 3.989/3 & 3.777/2 & 5.664/4 & 6.14/5 & **2.78/1** \\  & FE\({}_{\text{mean}}\) & 32.12/3 & 36.15/5 & 34.88/4 & 31.23/2 & **27.92/1** \\  & FE\({}_{\text{std}}\) & 4.674/3 & 4.865/4 & 2.813/2 & 5.241/5 & **2.722/1** \\  & FE\({}_{\text{mean}}\) & 39.30/4 & 40.66/5 & 39.20/3 & 36.41/2 & **33.42/1** \\  & FE\({}_{\text{std}}\) & 3.965/3 & **3.634/1** & 3.785/2 & 5.292/4 & 5.320/5 \\  & FE\({}_{\text{mean}}\) & 45.76/4 & 46.22/5 & 44.12/3 & 41.34/2 & **35.22/1** \\  & FE\({}_{\text{std}}\) & 3.456/3 & 2.973/2 & **1.894/1** & 5.157/5 & 3.49/4 \\  & FE\({}_{\text{mean}}\) & 117.66/5 & 115.67/4 & 105.27/3 & 97.77/2 & **89.27/1** \\  & FE\({}_{\text{std}}\) & 19.11/5 & 118.12 & 17.55/4 & 14.44/3 & **9.99/1** \\  & FE\({}_{\text{mean}}\) & Failed/5 & Failed/5 & Failed/5 & 140.42/2 & **116.67/1** \\  & FE\({}_{\text{std}}\) & Failed/5 & Failed/5 & Failed/5 & 75.66/2 & **30.63/1** \\  & FE\({}_{\text{mean}}\) & 300.4/3 & 534.6/4 & 540.4/5 & **167.1/1** & 199.2/2 \\  & FE\({}_{\text{std}}\) & 119.1/3 & 147.0/4 & 159.4/5 & **55.21/1** & 88.89/2 \\  & FE\({}_{\text{mean}}\) & 37.50/3 & 37.62/2 & 38.20/4 & 39.34/5 & **36.58/1** \\  & FE\({}_{\text{std}}\) & 3.50/4 & 3.46/3 & 3.29/2 & 3.64/5 & **2.56/1** \\  & FE\({}_{\text{mean}}\) & 107.03/4 & 105.13/3 & 103.67/2 & 114.1/5 & **101.16/1** \\  & FE\({}_{\text{std}}\) & 44.50/2 & 50.39/5 & 48.26/3 & 49.30/4 & **43.43/1** \\ \hline \hline \end{tabular}
\end{table}