# Kriging-Based Variable Screening Method for Aircraft Optimization Problems with Expensive Functions
## 1 Introduction  
Solving expensive optimization problems with high-fidelity CFD models and many variables usually requires a large computational cost. This lies in that the sample size required to find the optimal solution in the search space usually grows exponentially with the number of optimization variables [1]. A possible solution concerns replacing high-fidelity simulation models with metamodels to reduce the computation burden. A metamodel is an approximate model constructed by sampling points from high-fidelity simulation and fitting the objective function and constraints. The optimization process is then conducted on the approximate model to identify the design point. This process is repeated until the best design is obtained. Kriging metamodels, which model the problem functions by using stochastic processes, excel at capturing nonlinear and multimodal relationships inengineering systems [2]. However, it is difficult to construct effective metamodels for the method of Kriging, especially for high-dimensional optimization problems. Therefore, it requires reducing the dimensionality of high-dimensional optimization problems. Dimensionality reduction (DR) methods transform high-dimensional data into meaningful lower-dimensional representations [3]. The principal component analysis (PCA) [4] method is a linear method widely employed for dimensionality reduction, which describes the original data in a lower dimension and fits the variance of the original data as closely as possible. Other DR methods include Isomap [5], Kernel PCA [6], Maximum Variance Unfolding (MVU) [7], Local Linear Embedding (LLE) [8], and so on. Such DR techniques fall in the class of convex methods where the cost function must be strictly convex [3]. In contrast, we cannot ensure that the objective function has a unique minimum for a general engineering problem. While these DR methods are powerful tools for data compression and feature extraction, they are generally ineffective in reducing the computational complexity of high-dimensional black-box optimization problems. Variable screening identifies important inputs and prunes less significant variables or noises to reduce the dimensionality of the problems. The screening process is usually carried out via sampling and analyzing the sampling results. To distinguish the active inputs of a certain model, sensitivity analysis (SA), including both local and global ones, is widely employed. Local sensitivity analysis refers to the local variability of the output relative to the input variable at a certain point, which is actually a partial derivative. Global Sensitivity Analysis (GSA) consists of a series of mathematical methods for studying how changes in the output of a numerical model depend on the changes in its inputs. [PERSON] [9] proposed a variance-based technique utilizing Sobol variance decomposition [10], which turned out to be a valid GSA approach. However, the sample size of variance-based methods is large, so the number of evaluations required for such methods is computationally taxing for complex models. Thus, the sensitivity analysis based on variance is often performed by using a Gaussian process metamodel. [PERSON] [11] combines the quasi-optimized trajectory-based elementary effects (quasi-OTEE) method with the Kriging method to solve the high-dimensional SA problem. The first step is to use the quasi-OTEE SA to identify the active and inactive inputs of the problem. The Kriging-based SA is then leveraged to count the sensitivity indicators and sort the most active inputs more accurately. [PERSON] [12] provides a Matlab toolbox for solving GSA problems named SAFE (Sensitivity Analysis for Everybody), which contains a variety of GSA methods, such as the variance-based SA method. The Kriging-based SA method seems inaccurate in ranking the factors based on a small sample size, according to the results of some high-dimensional screening test problems.