# A Hybrid Adaptive Particle Swarm Optimization Algorithm for Enhanced Performance
## 2 PSO Algorithm
### Principle  
The PSO algorithm is a widely recognized method of population-based intelligent optimization. Similar to other algorithms in this category, it typically commences by randomly initializing a set of solutions, referred to as particles. The algorithm then iteratively updates these solutions, guiding the entire population toward enhanced fitness. The ultimate objective is to identify the optimal solution to the problem within a constrained number of iterations. If the PSO algorithm with \(N\) particles is employed to address a problem within a search space of dimension \(J\), the position and velocity components of the \(i\)-th particle (where \(1\leq i\leq N\)) at the \(n\)-th iteration can be expressed as \(X_{i}=(x_{i1},x_{i2},\ldots,x_{ij},\ldots,x_{ij})\) and \(V_{i}=(v_{i1},v_{i2},\ldots,v_{ij},\ldots,v_{ij})\), respectively. During each iteration, PSO employs specific formulas to update the velocity and position of each particle in each dimension component:  
\[v_{ij}^{t+1}=v_{ij}+c_{1}r_{1}(pBest_{ij}^{t}-x_{ij}^{t})+c_{2}r_{2}(gBest_{gj}^ {t}-x_{ij}^{t}), \tag{1}\]  
\[x_{ij}^{t+1}=x_{ij}^{t}+v_{ij}^{t+1} \tag{2}\]  
Equations (1) and (2) present the updated iterative formulas of the original PSO algorithm [1]. In these equations, \(c_{1}\) and \(c_{2}\) represent acceleration factors that are utilizedto modify the convergence speed, typically assuming a value of 2. \(r_{1}\) and \(r_{2}\) are random numbers distributed on [0, 1]; \(x^{\prime}_{ij}\) denotes the position component of the \(j\)-dimensional space of particle \(i\) in the \(t\)-th iteration; \(pBest^{t}_{ij}\) denotes the optimal position of an individual; \(gBest^{t}_{3j}\) denotes the optimal position of a group; after the completion of each iteration, the \(pBest\) position of each particle must be updated according to the following rules:  
\[p^{t+1}_{ij}= \begin{cases}x^{t+1}_{ij},&f(X^{t+1}_{ij})<f(p^{t}_{ij}),\\ p^{t}_{ij},&f(X^{t+1}_{ij})\geq f(p^{t}_{ij}).\end{cases}  \tag{3}\]  
where \(f(\bullet)\) represents the objective function used to determine the corresponding positional adaptation value. In many cases, it is necessary to confine particle velocities within a specific interval, denoted as \([-v_{max},v_{max}]\), to align with the practical requirements when employing the algorithm. However, researchers have observed that the initial version of the algorithm exhibited slow convergence rates when addressing certain problems, primarily due to its overly global search behavior. This limitation resulted in diminished algorithm performance. To address this issue, [PERSON] and [PERSON] [11] introduced an enhanced PSO algorithm incorporating inertia weights, which involves the addition of inertia weights to Equation (4):  
\[v^{t+1}_{ij}=\omega v_{ij}+c_{1}r_{1}\Big{(}pBest^{t}_{ij}-x^{t}_{ij}\Big{)}+ \ c_{2}r_{2}\Big{(}gBest^{t}_{jj}-x^{t}_{ij}\Big{)} \tag{4}\]  
The formula consists of three components: the first is termed the 'inertia component, which reflects the influence of the particle's inertia weights on its current velocity state, thereby preventing excessive oscillation during the search process. The inertia weights, denoted as \(\omega\), decrease linearly from 0.9 to 0.4 over the course of the iterations. This algorithm has been demonstrated to outperform alternative versions in numerous experiments and applications, and is therefore referred to as the standard PSO algorithm [26]. Unless otherwise specified, all references to the PSO algorithm in the subsequent sections pertain to the standard PSO algorithm. The second component is the \"individual cognition\" of the particle, which represents its learning regarding its own position. The third component is the \"social cognition\" of the particles, which reflects their learning of the optimal position within the group. The superposition of \"individual cognition\" and \"social cognition\" allows particles to engage in what is termed \"partial local search\", while the \"inertial part\" facilitates \"partial global search\".