## 4 Structural Optimization  
The displacement reducer proposed in this paper can be optimized by adjusting its geometric parameters. The objective function and parameter variables are determined using the analytical model calculated in Section 3. This paper utilizes the PSO algorithm for further structural optimization. The PSO algorithm is a heuristic optimization algorithm based on swarm intelligence. Its principle originates from simulating the behavior of biological groups, such as bird flocking foraging or fish schooling. The algorithm efficiently searches for the global optimal solution in the solution space by simulating the information sharing and collaboration mechanisms among individuals in the swarm. The core principle of PSO can be summarized as follows:  
Figure 10: Local sensitivity analysis of point and width: (**a**) The local sensitivity of point; (**b**) The local sensitivity of width. Figure 9: Analysis of the influence of different points in the y\(-\)coordinates on the reduction ratio: (**a**) point \(A\), (**b**) point \(B\), (**c**) point \(C\), (**d**) point \(D\), (**e**) point \(F\), (**f**) point \(G\), (**g**) point \(H\), (**h**) the relative position of points \(A\) and \(B\). Particle representation: In the PSO algorithm, each particle represents a candidate solution in the solution space. Each particle has two key attributes:  
* Position vector \(x_{i}(t)\): Represents the current position of the particle in the solution space, i.e., the current candidate solution;
* Velocity vector \(v_{i}(t)\): Represents the search direction and step size of the particle in the solution space that determine the direction and speed of the particle's next move. Velocity update mechanism: The velocity update of the particle is the core part of the PSO algorithm. The velocity update depends on the following three factors:
* Individual best solution \(p_{\text{best}}\): The best solution found by each particle during the search process, denoted as \(p_{\text{best}}\);
* Global best solution \(g_{\text{best}}\): The best solution found by the entire particle swarm during the search process, denoted as \(g_{\text{best}}\);
* Inertia weight \(w\): Used to balance global exploration and local exploitation. A larger inertia weight favors global exploration, while a smaller inertia weight helps local exploitation. The velocity update formula is as follows: \[v_{i}(t+1)=wv_{i}(t)+c_{1}r_{1}(p_{\text{best}}-x_{i}(t))+c_{2}r_{2}(g_{\text{ best}}-x_{i}(t))\] (10) where \(w\) is the inertia weight, which controls the inertial part of the particle's velocity. \(c_{1}\) and \(c_{2}\) are cognitive and social factors, respectively, used to adjust the particle's dependence on individual experience and group collaboration. Typically, \(c_{1}\) and \(c_{2}\) are set around 2. \(r_{1}\) and \(r_{2}\) are random numbers uniformly distributed in the range \([0,1]\), introduced to add randomness and avoid early convergence to local optima. Position update mechanism: The particle's position is adjusted according to the updated velocity. The position update formula is as follows: \[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1)\] (11) Through this formula, the particle moves to a new position in the solution space and continues to search for a better solution. Iterative optimization process: Through iterative updates of the particles' velocity and position, the swarm gradually converges towards the global optimal solution, ultimately obtaining the optimal solution that satisfies the optimization objectives. The PSO algorithm, with its simple structure, few parameters, fast convergence speed, and ease of implementation, has been widely applied in function optimization, engineering design, machine learning, and multi-objective optimization, among other fields. Its swarm collaboration mechanism and efficient global search capability make it one of the most effective tools for solving complex optimization problems.