## 2 Method Description
### Infill Criteria  
A popular and successful strategy is to combine different infill criteria. In this study, MSE + MSP is utilized to increase the surrogate model's precision. This infill criteria have some ability to explore and exploit.  
The MSE criteria take the point at which the objective function is maximized as the new sampling object (the objective function consists of the variance (\(s^{2}\)) of the probabilistic surrogate model), representing the exploration process, as in Equation (5):  
\[\begin{array}{ll}Max&F(x)=s^{2}\\ s.t.&g(x)=\overset{\rightarrow}{0}\end{array} \tag{5}\]  
The MSP criteria take the point at which the objective function is minimized as the new sampling object (the objective function is the predicted value of the surrogate model, which can be multiplied by \(-1\) when the actual problem is to find the maximum value), representing the exploitation process, as in Equation (6):  
\[\begin{array}{ll}Min&F(x)=\overset{g}{\rightarrow}\\ s.t.&g(x)=\overset{\rightarrow}{0}\end{array} \tag{6}\]  
where \(g(x)\) is the constraint function; '\(\rightarrow\)' refers to the vector.  
Figure 2: Latin square.  
In multi-objective optimization, the result usually exists as a Pareto set, and our goal is to obtain as true a Pareto Frontier as possible. Therefore, we need to select some Pareto solutions (points/new parameter combinations) to advance the Pareto Frontier in the right direction. It is worth noting that the points may form multiple clusters and that the shape of these clusters may be arbitrary. Filling according to clusters will significantly reduce the number of new samples.  
The MSE criteria and the MSP criteria play different roles in the optimization process, as the MSP criteria will usually provide the results we want. Therefore, in this paper, when constructing the objective functions for multi-objective optimization using the MSE criteria, a Pareto solution is selected from the Pareto set, and the point is used as the new sampling object. When constructing the objective functions for multi-objective optimization using the MSP criteria, the set \(X\) of points corresponding to the Pareto set is first clustered, and then some points (with the smallest weighting value) in partial clusters are selected as the next sampling points to contain more information in fewer samples and to quickly approach the true Pareto Frontier. In this paper, Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [54] is used in order to discover clusters with arbitrary shapes. The parameters necessary for the algorithm are the neighborhood radius (\(d\)) and the minimum number of points (\(p\)) within the radius. The process of the algorithm is as follows:  
1. All objects (points) in the initial set \(X\) are marked as 'unvisited';
2. Select an unvisited object \(x\) randomly, mark \(x\) as 'visited', and check whether the neighborhood of \(x\) contains at least \(p\) objects;
3. If not, then \(x\) is marked as a noise point. Otherwise, a new cluster \(C\) is created for \(x\), and all objects in the neighborhood of \(x\) are placed in the candidate set \(N\);
4. Add objects that do not belong to other clusters from \(N\) to \(C\) iteratively. In this process, for an object \(u\) from \(N\) marked as 'unvisited', mark it as 'visited' and check its neighborhood, and if the neighborhood of \(u\) contains at least \(p\) objects, then all objects in the neighborhood of \(u\) are added to \(N\). Continue adding objects to \(C\) until \(C\) cannot be extended (\(N\) is empty). Then, the generation of cluster \(C\) is complete;
5. Select an unvisited object at random from the remaining objects and repeat steps 2 and 3 until all objects have been visited.  
As the units of the parameters being optimized are usually different, this paper clusters by performing a dimensionless operation and maps the parameters to an interval with width 1.