## 3 Proposed Approach
### Step 4: Check the Terminal Condition  
Generally, there are two common ways to stop the sequential process. That is (1) the difference between the current optimal solution and the actual one achieves at an acceptable level and (2) all the computational resources are used up. In this work, these stopping criteria are adopted for different scenarios. For the numerical functions, because the actual optimal solution is known, the stopping criterion can be associated with this value to test the effectiveness of the proposed approach. Therefore, the stop condition is defined as \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Values** \\ \hline Population Size & 100 \\ Maximum generation & 100 \\ Crossover probability & 0.95 \\ Mutation probability & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameter setting of the genetic algorithm. Figure 2: The initial Kriging model of the illustrated example. \[\varepsilon_{r}=\left|\frac{\min(y_{k}(\mathbf{x}_{i}))-y^{r}}{y^{r}}\right|<\varepsilon _{g}\ i=1,2,\cdots,N \tag{19}\]  
where \(\min(y_{k}(\mathbf{x}_{i}))\) is the minimum value of the current sample set, \(y^{r}\) is the actual optimal solution, \(\varepsilon_{g}\) is a user-defined tolerance. Generally, the adaptive algorithm will confront the stricter test in the case of smaller tolerance. In this work, the value of \(\varepsilon_{g}\) is defined as 0.002 referring to [32]. However, for the engineering cases, the above stopping criterion for the numerical problem is impractical because the engineering problem is always a black-box problem. Thus the value of the actual optimal solution is unknown. Therefore, the sequential updating process terminates when the maximum iteration is reached, which can be expressed as  
\[k\geq K \tag{20}\]  
where \(k\) and \(K\) denote the current iteration and the maximum iteration, respectively. If the stopping criterion is satisfied, the sequential process will be terminated and the algorithm goes to Step 6. Otherwise, the proposed algorithm  
where  
\[\mathbf{p}_{ij}=\,Y_{ij}/\sum_{i=1}^{N}Y_{ij} \tag{24}\]  
If the value of \(\mathbf{p}_{ij}=0\), it indicates that the entropy of this tested point equals zero. In that case, a definition is given to compensate for the insufficiency of the initial assumption in Equation (23), which is defined as  
\[\lim_{\mathbf{p}_{ij}\to 0}\mathbf{p}_{ij}\ln(\mathbf{p}_{ij})=0 \tag{25}\]  
According to Equation (23), the degree of variation of each indicator can be ascertained. The indicator with a larger value of information entropy has a smaller degree of variation. Subsequently, the corresponding entropy weight should be small. As such, the entropy weight can be obtained by  
\[w_{j}=\frac{1-E(P_{j})}{\sum\limits_{j=1}^{m}\left(1-E(P_{j})\right)} \tag{26}\]  
According to Equation (26), the weight of each index can be determined adaptively. Besides, \(w_{j}\in[0,1]\) and \(\sum w_{j}=1\). Here we give a brief explanation of the proposed EW-LCB criterion. The term \(w_{1}\hat{g}(x)\) is used for local exploitation, which concerns the optimal value. On the other side, the term \(w_{2}\hat{g}(x)\exp((-1)^{r}r)\) focuses on global exploration, which pays more attention to the uncertainty of the Kriging model for the potential global optimal region. If \(w_{1}\ll w_{2}\exp((-1)^{r}r)\), it means the algorithm focuses more on global exploration. While \(w_{1}\gg w_{2}\exp((-1)^{r}r)\) means the algorithm focuses more on local exploitation. The factor \(\exp((-1)^{r}r)\) serves as the catalyst to help the optimization process out of a local optimization solution. However, this factor may decrease the convergence speed of the proposed algorithm because the weight of the exploration will dominate \(EWLCB(\mathbf{x})\) when the current optimization solution is repeated too many times. Finally, the point with the minimum value of \(EWLCB(\mathbf{x})\) is selected as the new update point. In this illustrated example, the weight parameters are \(w_{1}\)= 0.4961,\(w_{2}\exp((-1)^{r}r)=0.5039\) in the first iteration.