*3.3. The Particle Swarm Optimization–Butterfly Optimization Algorithm*

In the process of searching, it is simple for the PSO algorithm to fall into the local optimal solution. The main drawback of the BOA is its extended convergence time and significant oscillations during the process. To address the limitations of the PSO and BOA algorithms, the PSO-BOA algorithm incorporates the BOA search mechanism into the PSO algorithm. Specifically, the algorithm selects the search method based on comparing the generated random number with the predetermined switching probability '*p'*. To enhance the algorithm's ability to identify the global optimal value, the PSO-BOA algorithm integrates two strategies. On the one hand, the strategy randomizes the spatial position of individual particles during the local search process, which explores a diverse search space and optimizes the quality of individual particles. On the other hand, the strategy utilizes both local and global optimal particles to update the position and speed of particles in the global search process, enabling the algorithm to exploit the current best solutions and refine the search trajectory toward the global optimal value. Simultaneously, the PSO-BOA algorithm adjusts the sensory modality *c* and the inertia weight *ω* in the iteration process, which accelerates the convergence rate and improves the local search performance.
