# A Hybrid Adaptive Particle Swarm Optimization Algorithm for Enhanced Performance
## 3 Improved PSO Algorithm
### Elite Particle Learning Mechanism  
The particle swarm optimization algorithm classifies particles based on their fitness values, identifying those with lower fitness yet superior performance as the preferred subset, referred to as elite particles. Through continuous optimization and iteration, these elite particles play a critical role in the algorithm's pursuit of the global optimal solution, enabling a more precise search for the optimal position and thereby enhancing the algorithm's efficiency and solution quality. Within the particle swarm optimization framework, the primary position vectors involved in the iterative process include the historical optimal position and the global optimal position; both are essential to the velocity update mechanism of the particles. The essence of the elite particle learning mechanism lies in its ability to effectively leverage its own historical optimal position and the global optimal position to efficiently and accurately locate the global optimal solution during the search process. The velocity of the elite particle is calculated using the following formula:  
\[V1=\omega_{i}v_{i}^{d+1} \tag{18}\]  
\[V2=c_{1}r_{1}\Big{(}r_{1}\Big{(}pbt_{ij}^{t}-gbt_{gj}^{t}\Big{)}\Big{)}+\Big{(} gbt_{gj}^{t}-x_{ij}^{t}\Big{)} \tag{19}\]  
\[V3=c_{2}(1-r_{1})\times\Big{(}gbt_{gj}^{t}-x_{ij}^{t}\Big{)} \tag{20}\]  
\[sv_{ij}^{t+1}=V1+V2+V3 \tag{21}\]  
Among them, the velocity update of the elite particle is divided into three parts: the first part \(V1\) represents the previous velocity part; the second part \(V2\) represents the part of cross-learning between the individual optimal position \(pbt_{gj}^{t}\) and the global optimal position \(pbt_{gj}^{t}\); and the third part \(V3\) represents the part of social cognition based on the global optimal position. The velocity update of the elite particle is the sum of these three parts.  
The position update of the elite particles is calculated using Equation (22):  
\[x_{ij}^{t+1}=\omega_{i}x_{ij}^{t}+(1-\omega_{i})\times v_{ij}^{t+1} \tag{22}\]  
Elite particles enhance the iteration speed and optimization search accuracy of the algorithm by employing cross-learning and social learning strategies. These approaches prevent premature convergence of the particles, thereby avoiding failure to reach the global optimal solution. Particle cross-learning integrates the strengths of different particles to generate new search directions while continuously tracking and retaining their historical optimal positions. This adaptive adjustment of speed and position enhances the diversity of the particle swarm, facilitating movement beyond local optima to explore a broader search space. Conversely, social learning allows the algorithm to leverage the benefits of collective intelligence and information sharing, thereby accelerating convergence, mitigating oscillations due to excessive speed, and improving search efficiency and solution quality. The combination of these two learning strategies effectively balances the exploration and exploitation capabilities of the particles, significantly enhancing the performance of the particle swarm algorithm.