## 2 Related Works
### EHVI Infill Criterion for the Multi-Objective EGO Algorithm  
As discussed in Section 1, one of the most popular criteria designed for a multi-objective EGO (MOEGO) algorithm is the EHVI criterion, which is modified from HVI. Therefore, the HVI is described here before presenting the EHVI criterion. Considering the MOO problem with \(m>1\) objective functions \(f_{m}\): \(\mathbf{x}\mapsto\mathbb{R}^{m}\), the MOO problem can be briefly expressed as  
\[\left. \begin{array}{c}\underset{\mathbf{x}\in\mathbf{X}}{\text{Min}\text{ \emph{image}}}\mathbf{y}=\left\{f_{1}(\mathbf{x}),\ f_{2}(\mathbf{x}),\ \ldots,\ f_{m}(\mathbf{x})\right\}\\ \text{Subject to }\mathbf{x}\in[l,\ u]\end{array} \right\} \tag{8}\]  
where \(\mathbf{x}\subseteq\mathbb{R}^{d}\), \(\mathbf{y}\in\mathbf{Y}\subseteq\mathbb{R}^{m}\). \(I\) and \(\mathbf{u}\) are the lower and upper constraints, respectively, of the design variable vector \(\mathbf{x}\). \(d\) denotes the number of design variables, and \(m\) is the number of objectives. Assume the problem (8) to be optimized iteration by iteration with the EGO optimizer and take the set of Pareto front solutions (points) at the \(n\)-th iteration of problem (8) as \(\mathcal{P}_{n}=\left\{\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{n_{ dom}}\right\}\), where \(n_{dom}\) is the number of current Pareto front points; all the elements in \(\mathcal{P}_{n}\) are nondominated by each other, and can be mathematically expressed as  
\[\forall i,\ j\in\{1,\ 2,\ \cdots,\ n_{dom}\}\wedge(i\
eq j)\text{: }\Big{(}\mathbf{y}_{i}\
ot\prec\mathbf{y}_{j}\Big{)}\wedge\Big{(}\mathbf{y}_{ j}\
ot\prec\mathbf{y}_{i}\Big{)}\]  
in which \(a\
ot\prec b\) means \(a\) does not dominate \(b\), and on the contrary, \(a\prec b\) means \(a\) dominates \(b\). \(\mathbf{y}_{i}=\left\{y_{i}^{1},y_{i}^{2},\cdots,y_{i}^{m}\right\}\) denotes the coordinate of the \(i\)-th point in \(\mathcal{P}_{n}\). In order to compute the hypervolume used in the EHVI criterion, a reference point \(\mathcal{R}=\{r_{1},r_{2},\cdots,\ r_{m}\}\in\mathbb{R}^{m}\) is usually required to be defined by designers before the calculation, and then the hypervolume of the current Pareto front related to the reference point, \(\mathcal{H}(\mathcal{P}_{n},\ \mathcal{R})\), can be expressed as  
\[\mathcal{H}(\mathcal{P}_{n},\ \mathcal{R})=\int_{(\exists\mathbf{y}_{i}\in \mathcal{P}_{n})\prec\mathbf{y}\prec\mathcal{R}}\mathbf{dy} \tag{9}\]In Equation (9), one must notice that the hypervolume is equivalent to the region, which is dominated by at least one point of \(\mathcal{P}_{n}\) and itself dominates the reference point, as shown in Figure 1, shaded with blue oblique lines. Assume that the current predicted updating point is \(\mathbf{y}^{\prime}\), the HVI indicator \(I_{\mathcal{H}}(\mathbf{y}^{\prime},\ \mathcal{P}_{n})\) of the red area in Figure 1 can then be defined as:  
\[I_{\mathcal{H}}\big{(}\mathbf{y}^{\prime},\ \mathcal{P}_{n}\big{)}=\left\{ \begin{matrix}\mathcal{H}((\mathcal{P}_{n}\cup\{\mathbf{y}^{\prime}\}),\ \mathcal{R})\ -\ \mathcal{H}(\mathcal{P}_{n},\ \mathcal{R})&\
exists\mathbf{y}_{i}\in \mathcal{P}_{n}:\ \mathbf{y}_{i}\prec\mathbf{y}^{\prime}\\ 0&\text{Otherwise}\end{matrix} \right. \tag{10}\]  
It can be noted that only when \(\mathbf{y}^{\prime}\) falls in the \(\mathcal{P}_{n}\) nondominated region can the positive value of HVI be obtained. As expected, a large value of HVI reflects an improvement in the Pareto front solutions. Hence, the HVI, \(I_{\mathcal{H}}(\mathbf{y}^{\prime},\ \mathcal{P}_{n})\), can be taken as the infill criterion directly, and its further enhanced criterion of the HVI indicator, namely \(El_{\mathcal{H}}(\mathbf{x})\) of EHVI indicator, is proposed by [PERSON] et al.