# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
## 3 Case Studies of a Cooke Triplet System
### Case 2  
Since CODE V has its own built-in optimization module and it has been used as an industrial standard, a case study was carried out to show that the model presented here can further improve the CODE-V's optimization result. The CODE-V optimized values are shown in Tables 7 and 8. These were used as the starting point of the optimization process in Case 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Quantity** & **Optimized Value** & **CODE V Check** & **Deviation (\%)** \\ \hline DIS & 1.104833510 & 1.10483 & \(-\)0.000318\% \\ \(w_{1}\times\text{RMS1}+w_{2}\times\text{RMS2}+w_{3}\times\text{RMS3}\) & 0.031301775 & 0.031167 & \(-\)0.429832\% \\ RMS1 & 0.009467504 & 0.009824 & 3.761249\% \\ RMS2 & 0.043443469 & 0.04291 & \(-\)1.227962\% \\ RMS3 & 0.037875171 & 0.037719 & \(-\)0.412330\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Checking optimization results.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Quantity** & **Before Optimization (CODE V)** & **Optimization Result** \\ \hline DIS & 1.24963 & 1.104833510 \\ \(w_{1}\times\text{RMS1}+w_{2}\times\text{RMS2}+w_{3}\times\text{RMS3}\) & 0.03306 & 0.031301775 \\ RMS1 & 0.00856 & 0.009467504 \\ RMS2 & 0.04649 & 0.043443469 \\ RMS3 & 0.04062 & 0.037875171 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of results before and after optimization.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Initial Value (Unit: mm)** &
\begin{tabular}{c} **Optimized Value** \\ **(Unit: mm)** \\ \end{tabular} \\ \hline S1 & 21.48138 & 21.65449 \\ S2 & \(-\)124.1 & \(-\)124.40895 \\ S3 & \(-\)19.1 & \(-\)19.28859 \\ S4 & 22 & 22.14426 \\ S5 & 328.9 & 325.61100 \\ S6 & \(-\)16.7 & \(-\)16.74972 \\ D2 & 2 & 2.01141 \\ D3 & 5.26 & 5.20719 \\ D4 & 1.25 & 1.25230 \\ D5 & 4.69 & 4.73700 \\ D6 & 2.25 & 2.25106 \\ D7 & 43.0504842168944 & 42.95710 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of design parameters before and after optimization (Case 1).  
The other settings were all the same as Case 1. Based on the prediction of the Kriging surrogate model for the testing data, the Correlation Coefficient of the prediction results were all greater than 0.971, and the Relative Errors were less than 3% except RMS1, which was 5.6%.  
Multi-objective optimization was conducted using the NSGA-III algorithm, with the setting of 1000 population and 2000 evolutionary generations. Figure 8 shows the final Pareto frontier solution set with the initial state.  
As seen from Figure 8, although CODE-V has optimized its output, the model presented here can still further improve the optimization design. If the DIS value is chosen  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (Unit: mm)** \\ \hline S1 & 18.9211 \\ S2 & \(-\)55.9799 \\ S3 & \(-\)17.2447 \\ S4 & 18.3846 \\ S5 & \(-\)105.9429 \\ S6 & \(-\)15.2416 \\ D2 & 2 \\ D3 & 4.5035 \\ D4 & 1.25 \\ D5 & 6.675 \\ D6 & 2.25 \\ D7 & 41.5769 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Initial design parameters of Cooke triplet (Case 2).  
Figure 8: Comparison of the Pareto solution set and the initial state.  
as 0.63 as an acceptable value, the final optimization solution can be obtained from the solution set. The design parameters and targeted values before and after the optimization are listed in Tables 9 and 10.  
The optimized solution further improved the performance of the Cooke triplet, with a 3.53% reduction in weighted RMS and a 4.33% reduction in the DIS.