# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
## 2 Methodology
### DOE Method  
For optical imaging system design, the relationships among input design parameters and imaging quality are very complex. It would be prohibitively time-consuming to perform all the possible computer experiments in order to comprehend these relationships or find the optimal design. The statistical design of experiments is a technique that can be used to design a limited number of samples that could reflect the design space information.  
For the conventional optical imaging system, the number of design parameters is in the range of \(10^{1}\) to \(10^{2}\). Among the experimental design methods for computer experiments discussed in Section 1.3, the Latin hypercube design was applied in this model. As a modern and popular method for space-filling experimental design, LHS is a type of stratified Monte Carlo (MC), which allows the experimental designer total freedom in selecting the number of designs to run (as long as it is greater than the number of parameters). The Latin hypercube design is suitable for computer experiments with considerably large dimensions of the design space and has the advantage that the number of samples is not limited by the number of design parameters. Its operation process is simple and flexible and meets the requirement of reducing the sample scale in the case of a large number of design parameters. At present, the Latin hypercube design has become particularly popular among strategies for computer experiments [98].  
In view of the above-mentioned advantages, LHS was chosen as the DOE algorithm for the computer experiment of the optical imaging system. The Latin hypercube design requires the designer to specify the number of parameters and their ranges, as well as the number of sample points to run. Assuming that the dimension of the design space is \(n\), the number of sample points to be extracted is \(ns\), and the value range in a certain dimension  
Figure 1: Process steps.  
is \(x\in[l_{i},u_{i}](i=1,2,\ldots,n)\), where, \(l_{i}\) is the lower limit for the \(i\)-th parameter, and \(u_{i}\) is the upper limit for the \(i\)-th parameter. The main steps of the LHS experimental design are as follows:  
1. Give the scale of sampling \(ns\).
2. Divide the value range \([l_{i},u_{i}]\) of each dimension parameter \(x_{i}\) into \(ns\) intervals equally, then the design space can be divided into \(ns\times n\) sub-areas.
3. Randomly generate a matrix \(X\) with the order of \(ns\times n\). Each column of this matrix is a random arrangement from 1 to \(ns\) (elements are \(X_{i,j},\ i=1\cdots ns,j=1\cdots n\), which are random integers in the range from 1 to \(ns\)). The matrix \(X\) is called a Latin hypercube.
4. Each row of the matrix \(X\) corresponds to a selected small hypercube, which is a sample point. The normalized value of the \(i\)-th sample point for the \(j\)-th parameter can be calculated as: \(x_{i,j}=\big{(}X_{i,j}-0.5\big{)}/ns\).  
The actual value of the parameter for each sample point can be obtained by mapping \(x_{i,j}\) into the design space considering the actual range.  
Figure 2 shows the results of extracting 200 sample points in a two-dimensional space (each dimension range is [0, 1]) and 500 sample points in a three-dimensional space (each dimension range is [0, 1]) using the LHS method. The LHS method can guarantee that the number of projections of samples in each dimension of the design parameters is equal to the number of samples, and the projections have uniform distribution in each dimension.