## 2 Preliminaries
### Pso  
PSO is an optimization technique modeled after the collective behavior exhibited by social organisms, such as bird flocking and fish schooling, which was proposed in 1995 by [PERSON] and [PERSON]. This method leverages the collective intelligence of a population of entities, termed particles, to search for optimal solutions in a problem domain. In the PSO paradigm, particles traverse the search space, adjusting their trajectories based on their own best-known positions and the best-known position of the swarm. This collaborative search mechanism enables the identification of optimal solutions by capitalizing on both individual and collective experiences [10,11].  
Each particle within the swarm maintains an archive of its best-encountered solution, denoted as \(p_{\text{best}}\), and the swarm collectively maintains the best solution found thus far, referred to as \(g_{\text{best}}\). The velocity and position updates for each particle are steered by the following equations:  
\[v_{i}^{t+1}=w\cdot v_{i}^{t}+c_{1}\cdot r_{1}\cdot(p_{\text{best},i}-x_{i}^{t}) +c_{2}\cdot r_{2}\cdot(g_{\text{best}}-x_{i}^{t}) \tag{2}\]  
\[x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1} \tag{3}\]  
Here, \(v_{i}^{t}\) and \(x_{i}^{t}\) represent the velocity and position of particle \(i\) at iteration \(t\), respectively. The terms \(c_{1}\) and \(c_{2}\) are positive constants that determine the influence of the personal best and global best positions on the velocity update. \(r_{1}\) and \(r_{2}\) are random numbers uniformly distributed between 0 and 1, which introduce stochastic elements to the search process. The inertia weight \(w\) controls the impact of the previous velocity on the current velocity, thereby balancing exploration and exploitation in the search space.  
The elegance of PSO lies in its simplicity and adaptability, as it does not rely on the gradient of the objective function, making it a robust choice for non-linear and non-convex optimization scenarios [12]. Its capacity for swift convergence in expansive, multi-dimensional search spaces is particularly valuable for complex optimization tasks. In the context of SLAM, where the goal is often to find a single, globally optimal solution for the robot's pose and landmark positions, PSO's propensity for local convergence is less of a drawback. Instead, its ability to efficiently navigate the solution space and mitigate particle degradation makes PSO an advantageous choice for enhancing the performance of SLAM algorithms like FastSLAM.  
Moreover, the diversity of solutions provided by MOO can offer additional flexibility. For instance, different solutions on the Pareto front can be selected based on specific mission requirements or changing environmental conditions, allowing for greater adaptability and  
Figure 1: The Pareto front and dominance relationships.  
robustness in real-world SLAM applications. This flexibility is crucial for SLAM systems that must operate in diverse and unpredictable environments, where the ability to adapt to new conditions quickly is essential. Therefore, the integration of PSO with multi-objective optimization is particularly beneficial for SLAM, as it combines the efficient search capabilities of PSO with the comprehensive solution set provided by MOO. This synergy enables SLAM systems to not only find optimal solutions, but also to maintain a diverse set of good solutions that can be adapted to various scenarios.