## 4 MSIMP Based on EMOEAs
### Surrogate Model for EMOEAs  
In recent years, researchers at home and abroad have proposed a variety of surrogate model expensive multi-objective evolutionary algorithms, including EMOEAs based on multi-surrogate co-optimization models, transfer learning surrogate models, parallel surrogate models, and multi-objective Bayesian optimization, which have shown better performance in solving optimal MSIMP problems. Table 4 summarizes the main ideas, advantages, and disadvantages of various surrogate model expensive multi-objective evolutionary algorithms. Surrogate model expensive multi-objective evolutionary algorithms (SM-EMOEAs) have the following advantages with regard to solving the MSIMP problem. 1. They reduce the computational cost and improve the optimization efficiency: The surrogate model replaces the real objective function for evaluation and predicts the objective value of the candidate solution, allowing the evolutionary algorithm to efficiently generate and assess new individuals in each generation. This enables the algorithm to carry out more evaluations and optimizations, substantially reducing the large computational overhead within the limited time and resource constraints, thus accelerating the speed of convergence. The model maintains high computational efficiency when dealing with high-dimensional and complex problems, overcoming the computational bottlenecks of traditional optimization methods when dealing with high-dimensional optimal MSIMP problems. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Types** & **Surrogate Model** & **Main Ideas** & **Advantages and Disadvantages** \\ \hline \multirow{8}{*}{ANN} & \multirow{8}{*}{Deep Learning} & Complex objective functions are fitted through multiple hidden layers and nonlinear activation functions. & **Abvantages:** it can deal with highly complex and nonlinear problems and has strong adaptability. & **Abvantages:** it causes a large amount of data for training and complex parameter adjustment. \\ \cline{2-4}  & & \multirow{4}{*}{Deep Learning} & By increasing the depth of the network, it is able to handle more complex pattern recognition and prediction tasks. & **Advantages:** it adjusts the network architecture and loss function to adapt to different types of optimization problems. \\ \cline{2-4}  & & & **Disadvantages:** it has high data requirements and a need to label the data before training. \\ \cline{2-4}  & & \multirow{4}{*}{Deep Learning} & By constructing a decision tree, the dataset is split into regions and a simple regression is performed within each region. & **Advantages:** it has a good processing effect on nonlinear and complex data sets and is suitable for processing large-scale data. \\ \cline{2-4}  & & & **Disadvantages:** Making local decisions at each node cannot capture the global optimization solution. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._2. They enhance the diversity and quality of the solutions: Through efficient predictions and uncertainty steering provided by surrogate models, the algorithm can effectively balance exploration and exploitation during the evaluation process based on prediction uncertainty. The algorithm tends to prioritize exploring candidate solution areas with higher uncertainty, preventing premature convergence to local optima. For areas with lower uncertainty, the focus shifts to exploitation, further refining the existing solutions. The uncertainty steering encourages the generation of more diverse candidate solutions, thereby promoting population diversity. Additionally, multi-satellite imaging mission planning often involves multiple conflicting objectives, such as imaging coverage, image quality, and task completion time. By incorporating surrogate models with uncertainty guidance, the algorithm achieves balanced optimization across multiple objective functions, thereby improving solution quality. 3.