# A Hybrid Adaptive Particle Swarm Optimization Algorithm for Enhanced Performance
## 3 Improved PSO Algorithm
### Complex Chaotic Mapping Initialization Population  
In the traditional framework of particle swarm optimization algorithms, the initialization of particle populations typically relies on random methods. However, this random initialization can sometimes result in a less uniform distribution of the population within the solution space, increasing the risk of the algorithm converging to a local optimum. This, in turn, negatively impacts both the search efficiency and solution accuracy of the algorithm. To address this issue, the diversity of the population can be enhanced through chaotic mapping. The fundamental principle involves employing specific mapping techniques of chaotic sequences to translate the optimization variables into chaotic variable space, thereby leveraging the characteristics of chaos to significantly enhance population diversity. Several common chaotic mapping expressions are shown in Table 1.  
Different chaotic mappings exhibit distinct properties and exert varying effects on optimization algorithms. Logistic mappings enhance space traversal [27]; however, they may lead to aggregation or spatial white space. In contrast, Sine mappings possess a simpler structure and higher efficiency, which can enhance search accuracy and optimization performance. Tent mappings contribute to a more uniform distribution of particles, although later iterations may transition into a periodic sequence. Cubic and Chebyshev mappings are highly sensitive to initial values, resulting in uneven distribution across the mapping region. Given the numerous limitations associated with traditional chaotic mapping methods, researchers are increasingly favoring optimized chaotic mapping techniques for initializing population-based intelligence optimization algorithms. Among the various optimization schemes, Logistic mapping and Sine mapping are two extensively studied \begin{table}
\begin{tabular}{l l l} \hline
**Mapping Type** & **Formula** & **Range** \\ \hline Logistic & \(x_{k+1}=\gamma x_{k}(1-x_{k}),\gamma\in(0,4)\) & \([0,1]\) \\ Cubic & \(x_{k+1}=\rho x_{k}(1+x_{k}^{2})\) & \([0,1]\) \\ Chebyshev & \(x_{k+1}=cos(\alpha\frac{\alpha}{cos(x^{k})}),\alpha=\frac{1}{2}\) & \([-1,1]\) \\ Tent & \(x_{k+1}=\begin{cases}\frac{x^{2}}{\pi},\quad\quad x^{k}<\alpha\\ \frac{1-x^{2}}{1-\pi},\quad x^{k}\geq\alpha\\ \end{cases}\) & \([0,1]\) \\ Sine & \(x_{k+1}=asin(\pi x),\alpha\in[0,\,1]\) & \([0,1]\) \\ Sinusoidal & \(x_{k+1}=ax_{k}^{2}sin(\pi x),\alpha\in[0,\,4]\) & \([0,1]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Common chaotic mappings. chaotic mapping methods. The composite chaotic mapping that integrates Sine mapping and Logistic mapping offers significant advantages in terms of spatial traversal, population diversity, and distribution uniformity, while also maintaining a simple structure and high computational efficiency [28]. Consequently, we employ a composite chaotic mapping method that combines Sine mapping and Logistic mapping to initialize the population, thereby effectively enhancing the search efficiency and solution accuracy. The specific definition formula and initial value expression for the composite chaotic mapping are presented below:  
\[y_{i}^{d}=\left(\sin\left(\pi y_{i-1}^{d}\right)\right)^{aln\left(4 by_{i-1}^{ d}\left(1-y_{i-1}^{d}\right)d+c\right)} \tag{5}\]  
\[x_{i}^{d}=lb+y_{i}^{d}\times\left(ub-lb\right) \tag{6}\]  
where \(a\), \(b\), \(c\) are denoted as the control parameters with the range of values \(0\leq a\leq 1\), \(0\leq b\leq 5\), \(1.5\leq c\leq 5\), respectively; \(y_{i}^{d}\) and \(x_{i}^{d}\) denote the random number and the initial value of particle \(i\) in the \(d\)-th dimension space of the initial position, respectively; \(lb\) and \(ub\) denote the lower bound and upper bound of the range of values; and the range of values of the particle is \([lb,ub]\). To assess the feasibility of the population initialized using composite chaotic mapping, we compare it to a population initialized with a pseudo-random sequence.