# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
## 4 Conclusions  
An optimization model based on a surrogate model and a multi-objective optimization algorithm for an optical imaging system was established in this paper. The use of a surrogate model can significantly reduce the calculation cost but still keep a high level of accuracy, especially when the design space has a large dimension. Another advantage of this model is the ability to optimize multiple objectives simultaneously during the optimization process. This is achieved by using a multi-objective optimization algorithm. With the surrogate model and the multi-objective optimization algorithm, this model can significantly improve the efficiency of optical design.  
Two case studies of optimizing a Cooke triplet optical system were carried out with twelve design parameters and two optimization objectives:  
Case 1 showed that the optimized result from the model significantly improved the imaging quality of the initial design, with a reduction of 5.32% in RMS and 11.59% in DIS. Further verification conducted using CODE-V showed that the deviation from an actual calculation was less than 0.5%.  
Case 2 used an optimized result from CODE-V as the starting point and showed that the optimization from the model presented further reduced the weighted RMS by 3.53% and the DIS by 4.33%.  
As a result, the model presented in this paper is suitable for the optimization of optical system design, and it can further improve the optimization results from CODE-V. It has great potential to be used in the design process of complex high-precision optical systems.  
Writing--original draft preparation, L.S.; writing--review and editing, L.S.; visualization, W.Z., W.L. and Y.Z.; supervision, H.L.; project administration, C.D. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Quantity** & **Before Optimization** & **After Optimization** \\ \hline DIS & 0.65474 & 0.6264 \\ \(w_{1}\times\) RMS1 + \(w_{2}\times\) RMS2 + \(w_{3}\times\) RMS3 & 0.008984 & 0.008667 \\ RMS1 & 0.005349 & 0.0048218 \\ RMS2 & 0.010732 & 0.0091484 \\ RMS3 & 0.010352 & 0.011481 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison of key parameters before and after optimization.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Value (Unit: mm)** & **Optimized Value (Unit: mm)** \\ \hline S1 & 18.9211 & 19.051388 \\ S2 & \(-\)55.9799 & \(-\)55.586519 \\ S3 & \(-\)17.2447 & \(-\)17.268053 \\ S4 & 18.3846 & 18.303013 \\ S5 & \(-\)105.9429 & \(-\)105.810608 \\ S6 & \(-\)15.2416 & \(-\)15.151970 \\ D2 & 2 & 1.996235 \\ D3 & 4.5035 & 4.469327 \\ D4 & 1.25 & 1.254227 \\ D5 & 6.675 & 6.649926 \\ D6 & 2.25 & 2.252321 \\ D7 & 41.5769 & 41.567974 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison of design parameters before and after optimization (Case 2).  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.  
**Data Availability Statement:** The study did not report any data.  
**Conflicts of Interest:** The authors declare no conflict of interest.