## 3 PSO Based on ABC Search (F-WAPSO)  
### Improved PSO and Its Convergence Analysis  
#### 3.1.1 Improved PSO (WAPSO)  
PSO is a swarm intelligence evolutionary algorithm that simulates the foraging behavior of birds. The basic idea of PSO is to abstract a bird into a \"particle\" without mass or volume and is equivalent to a solution to a problem. The solvable group is equivalent to a flock, and the nearest bird to the food is equivalent to the optimal solution in each generation of the solvable group. The position of the food corresponds to the global optimal solution. During the process of searching for food, birds in flight have their own flight position and velocity. Each bird can obtain the flight status information of other birds in the population and can adjust its current flight velocity and direction according to this information, so as to reach the desired position [18]. The classical particle swarm velocity update formula has three parts. The first part is the current particle velocity, which indicates the current flight state of the particle. The second part is the difference between the current position of the particle and the historical optimal position of the population, and it reflects the cooperation and knowledge-sharing among particles. The third part is the difference between the current position of the particle and its historical optimal position, indicating that the action of the particle comes from its own experience. These three parts work together to determine the searching ability of the particle. The first part endows the current particle with vigor, expands the search space, and prevents local convergence. The second part and the third part are the information exchange and cognition of the particle population respectively, and the global optimal solution is sought by constantly modifying the velocity and direction of the particle. The velocity and position update equations of PSO with inertia weights introduced by [PERSON] and [PERSON] are shown in Equations (4) and (5) [19]:  
\[v_{ij}(\tau+1)=\omega v_{ij}(\tau)+\phi_{1j}(\tau)(gbest_{j}(\tau)-x_{ij}(\tau ))+\phi_{2j}(\tau)(pbest_{ij}(\tau)-x_{ij}(\tau)) \tag{4}\]  
\[x_{ij}(\tau+1)=x_{ij}(\tau)+v_{ij}(\tau+1) \tag{5}\]  
where \(i\) represents the total number of particles (\(i\) = 1, 2,..., \(s\)); \(j\) represents the dimensions (\(j\) = 1, 2,..., \(n\)); \(\phi_{1j}(\tau)=c_{1}\tau_{1j}(\tau)\), \(\phi_{2j}(\tau)=c_{2}\tau_{2j}(\tau)\), where \(c_{1}\) and \(c_{2}\) are acceleration factors; \(r_{1j}(\tau),\;r_{2j}(\tau)\in(0,1)\) are random numbers; \(\omega\) is the inertial weight, i.e., the weight that maintains the current velocity of the particle; \(x_{ij}(\tau)\) is the position of particle \(i\) in the \(j\)th dimension at time \(\tau\); \(v_{ij}(\tau)\) is the velocity of particle \(i\) in the \(j\)th dimension at time \(\tau\); \(pbest_{ij}(\tau)\) is the individual optimal solution of particle \(i\) in the \(j\)th dimension at time \(\tau\), representing the position with the best food source passed by individual birds in their own flight, and for PSO, it is the position with the best fitness value searched by individual particles in the iterative process; and \(gbest_{j}(\tau)\) is the global optimal solution in the \(j\)th dimension at time \(\tau\), representing the position with the best food source passed by birds in the course of flight, and for PSO, it is the position with the best fitness value searched by the whole population in the iterative process. According to Equation (4), \(c_{1}\) is multiplied by the difference between the position of the global historical optimal solution and the particle position, indicating the particle's ability to approach the global optimal position. That is, \(c_{1}\) controls the step size of the particle flying toward the global optimal solution. If \(c_{1}\) is too large, the particle is more likely to fly to the current global optimal position, but it is more likely to fall into the local optimal solution while accelerating the search process.