## 4 Model Lightweight Algorithm
### Pareto Solutions  
In practical applications, multi-objective intelligent optimization algorithms are often used to solve the constrained optimization problems of multi-objective functions. Commonly used multi-objective algorithms include multi-objective genetic algorithms, multi-objective simulated annealing algorithms, and multi-objective PSO algorithms. These three types of algorithms are based on different simulation and evolution strategies and transform multi-objective optimization problems into a series of single-objective optimization problems. The goal of the algorithm is to find the optimal solution of the objective  
Figure 9: The comparison of the fitting accuracy.  
function within an appropriate search space while considering the constraints. This process can be achieved by iteratively adjusting the parameters to obtain numerical approximations of the extreme values.  
PSO is an efficient parallel search algorithm that can find not only the optimal solution of a problem but also several good suboptimal solutions, thus providing multiple meaningful solutions for practical engineering problems. Based on this, a multi-objective PSO algorithm was used to optimize the response surface fitting equation system. According to the actual engineering needs, under a constant load, the importance ranking of design response objectives is as follows: upper structure engineering quantity > mid-span vertical deflection > axial force of the compressive chord air tube at the pier top > axial force of the stiffening truss > longitudinal internal force of the bridge deck bottom plate.  
The PSO program was written using MATLAB, and the operation process of the PSO algorithm is displayed in Figure 10.  
Figure 10: The operation process of the PSO algorithm.  
The specific optimization process is as follows.  
1. Initialize the particle population size as \(N=500\), the particle dimension as \(D=4\), the maximum number of iterations as \(T=50\), and the learning factors as \(c_{1}=c_{2}=1.49445\). Determine the particle position range by Table 1; the maximum particle velocity is \(v_{max}=1\) and the minimum particle velocity is \(v_{min}=-1\).
2. Initialize the population with random particle positions \(x\) and velocities \(v\) within the specified range, the individual best position \(p\) and the best value \(p_{\text{best}}\) for each particle, and the global best position \(g\) and best value \(g_{best}\) for the entire swarm.
3. Calculate the fitness function based on the response surface equation system and supplement the fitness weighting matrix to regulate the importance ranking of multi-objective parameter optimization.
4. Calculate the individual best value for each particle to screen out the global best value for the entire swarm.
5. Update all particle positions \(x\) and velocity values \(v\), handle the boundary conditions, and update the individual best position \(p\) and best value \(p_{best}\), as well as the global best position \(g\) and best value \(g_{best}\) for the entire swarm.
6. Check if the termination condition is met (i.e., reaching the maximum number of iterations); if so, terminate the search process and output the global optimization value; otherwise, continue with iteration optimization.  
The fitness evolution curve after optimization is shown in Figure 11.  
From Figure 11, it can be seen that the fitness function quickly converges after 13 iterations. At this point, the optimized results of \(R_{1}\)-\(R_{5}\) are, respectively, \(-777.910\), 65,492,00