## 3 ADGMPSO Algorithm
### Cat Chaos Initialization Population  
In evolutionary algorithms, whether the initial population distribution is uniform significantly affects the algorithm's solution accuracy and convergence speed [22]. Standard PSO initializes the population by generating random variables, which have a poor ergodicity and an uneven distribution of the initial individuals [23]. Chaotic mappings are often incorporated into the metaheuristic algorithms due to their advantages, such as a high ergodicity and randomness. [PERSON] and [PERSON] [24] were the first to apply chaotic systems to optics inspired optimization (OIO) algorithms to improve OIO's global convergence speed and accuracy. They proposed three methods for improving chaotic OIO by incorporating five chaotic mappings into the two components of OIO. Similarly, the bird swarm algorithm (BSA) is prone to premature convergence and can fall into local optimal solutions. Therefore, the literature [25] integrates chaotic mappings into the BSA to address these limitations. However, logistic chaotic mappings, widely used in metaheuristic algorithms, have certain drawbacks, such as a sensitivity to the initial values and a high probability of mapping point edges, resulting in a relatively uneven traversal. To overcome these limitations, [PERSON] et al. [26] proved that cat chaotic mappings have better chaotic properties and ergodicity than logistic mappings, making them a promising option for enhancing the optimization algorithms' exploration and exploitation capabilities. This study introduces the sat chaotic mapping in this context to enhance the diversity and uniformity of the initial population in the evolutionary algorithms. By leveraging the superior properties of the cat chaotic mapping, we can generate initial populations with a greater diversity and a more uniform distribution of individuals, thus enhancing thealgorithm's global search capabilities and improving its performance. Its chaotic sequence generation function is defined in Equation (4). \[ \begin{bmatrix}x_{n+1}\\ y_{n+1}\end{bmatrix} = \begin{bmatrix}1&1\\ 1&2\end{bmatrix}\begin{bmatrix}x_{n}\\ y_{n}\end{bmatrix} \ mod\ 1 \tag{4}\]  
To initialize the particle swarm population, a chaotic sequence matrix of the dimensions N \(\times\) D is generated using Equation (4), where N denotes the size of the population and D denotes the dimensionality of an individual. Following this, the chaotic sequences are mapped to the initial population of individuals using Equation (5). \[X_{ij}=lb_{j}+\left(ub_{j}-lb_{j}\right)\times y_{ij} \tag{5}\]  
In this context, \(lb_{j}\) denotes the minimum value of the \(j\)th dimensional search range, while the symbol \(ub_{j}\) indicates the \(j\)-dimensional search space's maximum value. Sine-Based Nonlinear Decreasing Inertia Weights and the Asynchronous Change Learning Factor Strategy  
The inertia weight, denoted by the symbol '\(\omega\)', is a crucial PSO parameter that improves the algorithm's accuracy and speed of convergence. Standard PSO utilizes a linear decreasing inertia weighting strategy. However, the solution search process for real-world problems is typically nonlinear [27]. Therefore, this paper uses a sine-based nonlinear decreasing inertia weights strategy to improve the inertia weights, as shown in Equation (6). \[\omega=\omega_{max}-\frac{t\times(\omega_{max}-\omega_{min})}{T}\times sin \frac{(t\times\pi)}{2T} \tag{6}\]  
The variable \(T\) represents the maximum number of iterations for the evolutionary process, while \(t\) indicates the current iteration number. Figure 3 shows the improved inertia weight map, where \(\omega\) is large at the beginning of the iterative evolution, which facilitates a global search in the solution space. Then, it decays rapidly. In the late iteration, the exact search is performed with a small \(\omega\). In addition to the inertia weights, the learning factors c\({}_{1}\) and c\({}_{2}\) are vital parameters that affect the performance of PSO.