## 2 Related Works  
### Traditional Single-Objective EGO Algorithm  
A study of EGO to be used for optimizations with multiple objectives. The traditional EGO algorithm is usually developed for optimization with a single objective. For the sake of completeness, a brief discussion of the traditional single-objective EGO algorithm [1] is presented in this section. To be the core components of the traditional EGO algorithm, the Kriging surrogate model and EI infill criterion ensure its high efficiency of EGO and global searching ability. Kriging was first proposed by [PERSON] [32] in 1951, and its formulation with \(d\) variables and \(n_{s}\) samples can be described as follows: let \(n_{s}\) sampled points be \(\mathbf{X}\mathbf{s}^{n_{s}}=\left\{\mathbf{x}^{1},\mathbf{x}^{2},\cdots,\mathbf{x}^{n_{s}}\right\}\), and their associated objective function values are \(\mathbf{Y}\mathbf{s}^{n_{s}}=\left\{\mathbf{y}^{1},\mathbf{y}^{2},\cdots,\mathbf{y}^{n_{s}}\right\}\), in which the \(i\)-th sample point and its objective value are \(\mathbf{x}^{i}=\left(x^{i}_{1},x^{i}_{2},\cdots,x^{i}_{d}\right)\), \(\mathbf{y}^{i}=\left(y^{i}_{1},y^{i}_{2},\cdots,y^{i}_{d}\right)\), respectively, for \(i=1,\ 2,\ \cdots,\ n_{s}\). \[Y(\mathbf{x}^{i})=\sum_{j=1}^{k}\beta_{j}f_{j}(\mathbf{x}^{i})+z(\mathbf{x}^{i}) \tag{1}\]  
where \(f_{j}(\mathbf{x}^{i})\) are \(k\) known regression models, and \(\beta_{j}\) are correlation coefficients of \(f_{j}(\mathbf{x}^{i})\), \(z(\mathbf{x}^{i})\) is a stochastic model and its mean is zero and variance is \(\delta_{z}{}^{2}\). The covariance between two design points \(\mathbf{x}^{i}\) and \(\mathbf{x}^{i}\) can be written as  
\[Cov\Big{[}z\Big{(}\mathbf{x}^{i}\Big{)},\ z\Big{(}\mathbf{x}^{i}\Big{)}\Big{]}=\delta _{z}{}^{2}\mathbf{R}\Big{[}R\Big{(}\mathbf{x}^{i},\ \mathbf{x}^{i}\Big{)}\Big{]} \tag{2}\]  
where \(i,\ j=1,\ 2,\ \cdots,\ n_{s}\); a symmetric matrix, \(\mathbf{R}\), is a correlation function that reflects the relationship between all the sample points; \(R\) is a correlation function defined by the user, and a widely used form of \(R\) can be expressed as [33]  
\[R\Big{(}\mathbf{x}^{i},\mathbf{x}^{i}\Big{)}=\prod_{k=1}^{d}esp\Big{(}-\theta_{k} \Big{|}x^{i}_{k}-x^{i}_{k}\Big{|}^{p_{k}}\Big{)} \tag{3}\]  
where \(\theta_{k}>0\), \(0\leq p_{k}\leq 2\) are two hyper parameters. Once the hyper parameters \(\theta_{k}\) and \(p_{k}\) are determined, the prediction \(\hat{y}(\mathbf{x})\) of \(Y\) at the location \(\mathbf{x}\) can be calculated by  
\[\hat{y}(\mathbf{x})=\hat{\beta}+\mathbf{r}^{T}(\mathbf{x})\mathbf{R}^{-1}(\mathbf{y}-\mathbf{1}\beta) \tag{4}\]  
in which the \(n\)-vector, \(\mathbf{r}(\mathbf{x})\), reflects the correlations between the predicted location \(\mathbf{x}\) and the sampled locations \(\mathbf{R}\big{(}\mathbf{x},\mathbf{x}^{i}\big{)}\), where \(j=1,\ 2,\ \cdots,\ n_{s}\); \(\mathbf{1}\) is an \(n\)-vector of ones; and an \(n\)-vector, \(\mathbf{y}\), is the value of accurate response; \(\beta\) is the mean of \(Y(\mathbf{x})\), and \(\hat{\beta}\) denotes the generalized least squares estimator of \(\beta\), which is defined as \(\hat{\beta}=\big{(}\mathbf{1}^{T}\mathbf{R}^{-1}\mathbf{1}\big{)}^{-1}\mathbf{1}^{T}\mathbf{R}^{-1} \mathbf{y}\)[33], and then, the mean squared error (MSE) of \(\hat{y}(\mathbf{x})\) can be written as  
\[\text{MSE}[\hat{y}(\mathbf{x})]=\delta_{Z}{}^{2}\Big{(}1-\mathbf{r}^{T}(\mathbf{x})\mathbf{R}^ {-1}\mathbf{r}(\mathbf{x})\Big{)} \tag{5}\]The hyper parameter \(\theta_{k}\) (see Equation (3)) can be determined by using the maximum likelihood estimation (MLE) approach with the limitation of \(\theta_{k}>0\)  
\[\text{Ln}(\text{Likelihood})=-\frac{1}{2}\Big{(}n\ \ln\left(\delta_{z} ^{2}\right)+\ln\lvert R\rvert\Big{)} \tag{6}\]  
where \(\delta_{z}^{2}=\big{(}\mathbf{y}-1\hat{\beta}\big{)}^{T}\mathbf{R}^{-1}\big{(} \mathbf{y}-1\hat{\beta}\big{)}/n\), and by solving Equation (6), the Kriging model can be constructed. The approximation \(\hat{y}(\mathbf{x})\) and MSE can be obtained by Equations (4) and (5).