## 1 Introduction  
Real-world engineering optimization problems often involve many conflicting objectives and some constraints [1]. A constrained multi-objective optimization problem (CMOP) with inequality constraints can be expressed as:  
\[ \begin{split}&\text{minimize }F(x)=(f_{1}(x),\ldots,f_{m}(x))^{T},\\ &\text{s.t. }g_{j}(x)\leq 0,\quad j\in\{1,2,\ldots,n\},\\ & x\in\mathbb{R}^{d}\end{split}  \tag{1}\]where \(x=\left[x_{1},x_{2},\ldots,x_{D}\right]^{T}\) is a d-dimensional decision variable, \(f_{1}(x),\ldots,f_{m}(x)\) is the objective function to be optimized, and \(g_{j}(x)\)is the first j inequality constraint. For CMOP, constraint violation (CV) can be computed in the following way:  
\[\text{CV}(x)=\sum_{j=1}^{n}\max(0,g_{j}(x)) \tag{2}\]  
A feasible solution refers to a solution whose CV value is less than or equal to 0. When a solution \(x^{*}\) outperforms solution \(x\) on all objectives, \(x^{*}\) dominates \(x\). If \(x^{*}\) is not dominated by other solutions, \(x^{*}\) is called a non-dominated solution. The set of objective vectors formed by non-dominated solutions is called the constrained Pareto front. The goal of CMOP is to find the constrained Pareto front (CPF). There are a large number of constrained multi-objective problems in engineering problems in the real world. Some of these problems require expensive computational costs when being solved, such as fuel cell vehicle design [2], wire manufacturing [3], and space system design [4]. These types of problem are called expensive constrained multi-objective problems. The ECMOP requires the algorithm to find the optimal solution under limited computational resources. However, the feasible domain is extremely narrow in MW1, which makes it difficult for the algorithm to find the feasible domain quickly. Meanwhile, the feasible domain of the problem is very broad for CF1 if the computational resources are heavily used in the constraint-value dominated search process, and the algorithm often falls into local convergence, making it difficult to approach the Pareto frontier. Therefore, the algorithm must use different search strategies for different feasible domains under different feasible domain conditions. In the case that the feasible domain is narrow and difficult to find, the algorithm needs to find the feasible domain with as few evaluations as possible, and in the case that there are already enough individuals in the feasible domain, it will not obtain good results if it pays too much attention to the convergence and feasibility. However, due to the limitation of computational resources, it is difficult for the algorithm to search for the optimal solution in a large decision space, requiring the algorithm to converge quickly to find the feasible domain. On the other hand, the complexity of CPF makes it difficult to explore the feasible domain with a limited number of solutions, which requires the algorithm to have good diversity. Evolutionary algorithms are widely used in constrained multi-objective problems due to their advantages, such as simple implementation and strong search ability [5]. Constrained multi-objective evolutionary algorithms require using a large number of function evaluations to search the objective space to ensure the good performance of the algorithm. However, for ECMOPs, where the number of available function evaluations is limited, traditional constrained multi-objective evolutionary algorithms are not adept at generating solutions that meet the requirements [6]. In order to solve such problems, many researchers have used surrogate models instead of real evaluation functions to produce suitable solutions within a limited number of evaluations; this method is called surrogate-assisted MOEAs (SA-MOEAs). SA-MOEAs have become a promising direction in recent years [7] that has received increasing attention. Commonly used surrogate models in recent years include artificial neural networks and support vector machines [8; 9], radial basis functions [10], and Gaussian regression (Kriging) [11].