# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
## 3 Case Studies of a Cooke Triplet System
### Case 1  
A Cooke triplet system that consists of three lenses was used as the optimization design subject, as shown in Figure 3. The geometric shape and the distance between the lens were selected as the design parameters. The optimization's objectives were to minimize the maximum field curvature (DIS) and the geometric spot diagram (RMS) of the Cooke triplet system. The front and back curvature, thickness, and spacing of each lens were chosen as the design parameters. In total, there were 12 of them (not including D1, which is the distance to the system's origin), as shown in Figure 4. The initial design parameters from which the optimization started are listed in Table 2.  
Figure 4: Schematic diagram of the design parameters of Cooke triplet (D for distance, S for radius of curvature). Figure 3: Schematic diagram of Cooke triplet. Since there are two objectives, minimizing DIS and RMS, this was a two-objective optimization. Here, DIS was treated as a single value, while the three RMS were combined into one by using weighting factors, as seen in Equations (14) and (15). The weighting factors \(w_{1}\), \(w_{2}\), and \(w_{3}\) used here were 0.3, 0.35, and 0.35, respectively. \[Aim1:\ min(DIS) \tag{14}\]  
\[Aim2:\ min\ (w_{1}\times RMS_{1}+w_{2}\times RMS_{2}+w_{3}\times RMS_{3}) \tag{15}\]  
The range of variation for each design parameter was set as \(\pm 1\%\) of the initial value. The LHS method was used to choose 1200 sample points within this 12-dimension design space. Figure 5 shows the projections of these sample points in a 2D space (S1 X S2) and a 3D space (D2 X D3 X D4). A commercial software, CODE-V [146], was used to carry out the calculation at these sample points using a ray-tracing-based method and to provide the DIS and RMS at these sample points. Of the sample results, 95% were randomly selected as the training data for the Kriging surrogate model, and the remaining 5% were used for testing. Table 3 shows the evaluation results of each target value based on the 5% testing samples. From the table, the Correlation Coefficients are all close to 1. The Relative Errors are less than 1% except RMS1, which is 5%. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **(Unit: mm)** \\ \hline S1 & 21.48138 \\ S2 & \(-124.1\) \\ S3 & \(-19.1\) \\ S4 & 22 \\ S5 & 328.9 \\ S6 & \(-16.7\) \\ D2 & 2 \\ D3 & 5.26 \\ D4 & 1.25 \\ D5 & 4.69 \\ D6 & 2.25 \\ D7 & 43.0504842168944 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial design parameters of Cooke triplet (Case 1). Figure 5: Projection of sample points in 2D and 3D spaces: (**a**) 2D (S1 X S2) space; (**b**) 3D (D2 X D3 X D4) space. Multi-objective optimization was conducted using the NSGA-III algorithm. The population number was set at 1000, and the evolutionary generation was set at 2000. Figure 6 shows the Pareto frontier at different evolution steps in the evolution process. The shape of the frontier tends to stabilize after about 100 generations (last is 2000). The final Pareto frontier solution set and the initial design are shown in the Figure 7. Since this was a multi-objective analysis, the final result was not unique but a set of non-dominated solutions. It is very obvious that the optimization process significantly reduced both the weighted RMS and DIS. The final optimal solution can be chosen from the solution set depending on the design focus and strategy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Evaluation Parameter** & **DIS** & **RMS1** & **RMS2** & **RMS3** \\ \hline Average Relative Error & 3.18682 \(\times\) 10\({}^{-6}\) & 0.0513925 & 0.00582908 & 0.006664 \\ Root-Mean-Square Error & 5.49953 \(\times\) 10\({}^{-6}\) & 0.00098156 & 0.000428076 & 0.0003938 \\ Correlation Coefficient & 1.00 & 0.99578 & 0.999379 & 0.998721 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation results of the trained surrogate model. Figure 6: Pareto frontier evolution process. Figure 7: Comparison of the Pareto solution set and its initial state.