# Kriging-Based Variable Screening Method for Aircraft Optimization Problems with Expensive Functions
## 3 Proposed KRG-VSM  
### Procedure of KRG-VSM  
Kriging models the function as a stochastic process. \[y(x)=F\beta+\varepsilon,\varepsilon\sim N\Big{(}0,\sigma^{2}R\Big{)} \tag{4}\]  
The regression model \(F\beta\) offers little advantage in applications, which is set to be a constant. Then, \(F\) would be a one-dimensional vector because the regression model has only a constant term. \(R\) denotes the stochastic-process correlations among the inputs. For two vectorial inputs \(\omega\) and \(x\), the correlation \(R(w,x)\) can be expressed as  
\[R(w,x)=\prod_{j=1}^{d}\exp\Big{(}-\theta_{j}\big{|}w_{j}-x_{j}\big{|}^{2}\Big{)} \tag{5}\]  
where \(\theta_{j}\geq 0\). \(d\) is the dimensionality of the problem. Note that the function \(R(w,x)\) should be selected based on the characteristics of the objective function. The correlation function (5), which has higher-order derivatives, is appropriate for a smooth response, e.g., the aircraft optimization problems. Correspondingly, this type of function is not suitable for fitting step-response-type black-box functions. Given the parameter \(\theta\), the maximum likelihood estimates of \(\beta\) and \(\sigma^{2}\) are  
\[ \begin{split}\widehat{\beta}&=\Big{(}1^{T}R^{-1} 1\Big{)}^{-1}\Big{(}1^{T}R^{-1}Y\Big{)}\\ \sigma^{2}&=\frac{(Y-1\widehat{\beta})^{T}R^{-1}(Y- 1\beta)}{n}\end{split}  \tag{6}\]  
The regression process of the Kriging metamodel is to find a set of correlation parameters \(\theta\) to maximize the log-likelihood  
\[-\frac{1}{2}\Big{(}n\ln\sigma^{2}+\ln\det R\Big{)} \tag{7}\]  
The variable screening process is correlated to the regression process of the Kriging metamodel. The flowchart of KRG-VSM is illustrated in Figure 1. It should be noted that, unlike the regression problem in Kriging modeling, the maximum likelihood function optimization in this variable screening process is a single-variable optimization problem, as shown in Figure 1. The independent variables for the first and second optimizations are \(\theta\) and \(\theta_{j}\), respectively. The genetic algorithm (GA) was employed to maximize the log-likelihood function because conventional linear search methods failed to locate the global optimum in this non-convex optimization problem. Step 1. \(N\) initial samples are produced throughout the entire design space leveraging LHS method. Step 2. Let \(S\) denote the set of parameters of inputs restrained to share a common value of \(\theta\) so that the optimization of the likelihood function is only about \(\theta\). All inputs are on the same footing because of the same scales of the factors. Set \(S=1,...,d\). Common \(\theta\) is set to be the upper limit at first. Step 3. GA optimization is performed to maximize the log-likelihood subject to \(\theta_{1}=\cdots=\theta_{d}=\theta\), and the maximum by \(L_{0}\). The objective function can be expressed as  
\[-\frac{1}{2}\Big{(}n\ln\sigma^{2}+\ln\det R\Big{)} \tag{8}\]  
where the log-likelihood can be calculated by combining Equations (5) and (6). The fitness tolerance is set as \(10\times 10^{-5}\). The variable \(\theta\) is bounded between \(-1\) and \(1\). The maximum number of generations is equal to the problem dimension \(d\), and the population size is fixed at 50 individuals. All optimization problems considered in this study are formulated as unconstrained optimization problems. The maximum number of executions for the GA is  
Figure 1: Flowchart of KRG-VSM. \(50*d\). If the value of common \(\theta\) is reduced but greater than \(10\times 10^{-6}\), the process skips to Step 4. If not, the screening process terminates. Step 4. For every \(j\) in \(S\), GA is used to maximize the log-likelihood subject to \(\theta_{1}=\ldots=\theta_{j-1}=\theta_{j+1}=\ldots=\theta_{d}=\theta\) and the maximum is denoted by \(L_{j}\). Step 5. Let \(j^{\star}\) denote the input producing the largest increase, \(L_{j}-L_{0}\), in the log-likelihood at Step 4. If \(L_{j^{\star}}-L_{0}>0\), the process turns to Step 6. Otherwise, the screening process terminates. Step 6.