## 2 Theoretical Basis
### MOPSO-h Algorithm and Applications  
#### 2.2.1 Algorithm Principles  
Multi-objective optimization algorithms are designed to handle multiple conflicting objective functions simultaneously. Their primary aim is to identify the Pareto optimal solution set, thereby providing decision-makers with multi-dimensional trade-off solutions. In contrast to conventional single-objective optimization frameworks, the central challenge lies in balancing the convergence, distribution, and diversity of solutions. Prominent algorithms such as non-dominated sorting genetic algorithm-II (NSGA-II), non-dominated sorting genetic algorithm-III (NSGA-III), multi-objective evolutionary algorithm based on decomposition (MOEA/D), and MOPSO [29; 30; 31; 32] employ advanced mechanisms like non-dominated sorting procedures, decomposition-based strategies, or indicator-based selection mechanisms to address intricate trade-offs. In the aerospace domain, multi-objective optimization finds extensive applications in aircraft aerodynamic optimization, trajectory optimization, and mission planning. This approach enables the coordination of conflicting requirements, such as fuel efficiency, penetration probability, and miss distance, ultimately delivering globally optimal solution clusters for complex aerospace systems. The MOPSO algorithm integrates the efficient search mechanism of PSO with a multi-objective optimization framework, enabling it to rapidly approximate the Pareto front via iterative swarm intelligence. It is particularly effective for solving high-dimensional, nonlinear optimization problems and excels in handling physical constraints, such as overload limits and power models, making it highly suitable for aircraft trajectory optimization. In each iteration, the particle population is characterized by their position, \(\mathbf{X}_{n}^{i}\in\mathbb{R}^{|\mathbf{X}|}\), and speed, \(\mathbf{V}_{n}^{i}\in\mathbb{R}^{|\mathbf{X}|}j=1,2,\ldots,I,n=1,2,\ldots,N\). At the \(i\)-th iteration, the objective vector \(J\big{(}\mathbf{X}_{n}^{i}\big{)}=\big{[}J_{1}\big{(}\mathbf{X}_{n}^{i}\big{)},J_{2} \big{(}\mathbf{X}_{n}^{i}\big{)},J_{3}\big{(}\mathbf{X}_{n}^{i}\big{)}\big{]}\) is computed for each particle. Subsequently, non-dominated sorting and diversity pruning are applied to generate a set of non-dominated solutions, \(\Big{\{}\mathbf{X}_{1}^{i},\mathbf{X}_{2}^{i},\ldots,\mathbf{X}_{N_{A}}^{i}\Big{\}}\)[33]. The global leader \(\mathbf{X}_{A}^{i}\sim\sum\limits_{a=1}^{A}P_{a}\geq p\) & \(\sum\limits_{a=1}^{A-1}P_{a}<p,A\in\{1,2,\ldots,N_{A}\}\) is then selected from the current population based on a roulette wheel probability \(P\), where the selection probability \(P_{a}\) is determined by the crowding distance \(CD\Big{(}\mathbf{X}_{A}^{i}\Big{)}\), as follows:  
\[P_{a}=\frac{CD\Big{(}\mathbf{X}_{a}^{i}\Big{)}}{\sum\limits_{b=1}^{N_{A}}CD\Big{(} \mathbf{X}_{b}^{i}\Big{)}}. \tag{14}\]  
Furthermore, the MOPSO algorithm employs the mechanism \(\mathbf{X}_{A}^{i}\) for updating the speed and position of each particle, as follows:  
\[ \begin{array}{l}\mathbf{V}_{n}^{i+1}=w^{i}\mathbf{V}_{n}^{i}+a_{1}r_{1}\Big{(}\mathbf{X}_ {B}^{i}-\mathbf{X}_{n}^{i}\Big{)}+a_{2}r_{2}\Big{(}\mathbf{X}_{A}^{i}-\mathbf{X}_{n}^{i} \Big{)}\\ \mathbf{X}_{n}^{i+1}=\mathbf{X}_{n}^{i}+\mathbf{V}_{n}^{i+1}\end{array} , \tag{15}\]  
where \(w^{i}=w_{\max}-(w_{\max}-w_{\min})i/I\) is the dynamic inertia weight and \(w_{\max}\) and \(w_{\min}\) are the upper and lower bounds of the dynamic inertia weight, respectively. At the same time, the individual historical best \(\mathbf{X}_{B}^{i}\) is updated based on the following calculations, and whether the iteration should be terminated is determined:  
\[\mathbf{X}_{B}^{i+1}=\left\{ \begin{array}{l}\mathbf{X}_{n}^{i+1},\text{if }\mathbf{X}_{n}^{i+1}<\mathbf{X}_{B}^{i}\text{ or incomparable}\\ \mathbf{X}_{B}^{i},\text{else}\end{array}