## 2 Theoretical Background and Problem Statement
### Improved Sparrow Search Algorithm (ISSA)
This process iterates, with continuous updates to the global best solution, until either convergence is reached or the maximum number of iterations is completed, at which point the best position and fitness score are generated as the final outputs. Although the traditional sparrow search algorithm (SSA) has demonstrated a commendable performance in various optimization problems through its emulation of natural sparrow foraging and predator evasion behaviors, the algorithm encounters challenges when addressing some complex optimization scenarios. In high-dimensional and multimodal problems in particular, traditional SSA can sometimes become trapped in local optima, and its convergence speed may be suboptimal [28; 29; 30; 31]. Moreover, balancing exploration and exploitation capabilities remains a key area for enhancement. To overcome these limitations, this research proposes an improved sparrow search algorithm (ISSA) that incorporates several key innovations to enhance the original framework:  
1. The good point set (GPS) is a strategy for generating initial solution sets characterized by high uniformity and low correlation. It is suitable for use in global optimization problems. This strategy involves selecting the smallest prime number \(p\) greater than \(2N+3\), coupled with the use of the cosine function, to create a point set. The specific formula for generating points is as follows: \[G_{ij}=i\times 2 cos\left(\frac{2\pi\times j}{p}\right)\mod 1,i=1,2,\ldots,M;j=1,2,\ldots,N\] (17) where \(i\) and \(j\) represent the indices of the points and dimensions, respectively, \(M\) denotes the total number of points, and \(N\) indicates the dimensions of the problem. Each point \(G_{ij}\) is subsequently mapped to the actual search space by scaling linearly to the respective dimension's upper \(ub_{j}\) and lower bounds \(lb_{j}\): \[Positons=lb_{j}+\left(ub_{j}-lb_{j}\right)\times G_{ij}\] (18) Thus, the good point set ensures the uniform distribution of the initial population in the parameter space, effectively enhancing the diversity of solutions and reducing clustering among them. This contributes to improving the global search capabilities and the efficiency of optimization algorithms. 2. By dynamically adjusting the proportion of discoverers and joiners, this improvement aims to balance exploration and exploitation capabilities. The proportional coefficient changes dynamically with the number of iterations, allowing for more explorers to  
Figure 2: The workflow of ISSA. investigate the solution space early in the search and increasing the number of joiners later on in order to refine potential solution areas already identified [27]. The calculation method for the proportional coefficient r is as follows:  
\[\text{r}=\text{b}\times\left(\tan\biggl{(}-\frac{\pi i}{4\times Max_{i}\text{ ter}}+\frac{\pi}{4}\biggr{)}\right)-\text{k}\times\text{rand} \tag{19}\]  
where \(i\) is the current iteration number, b and k are coefficients, and rand generates a random number within the [0, 1] interval. \(Max\_iter\) indicates the maximum number of iterations. This enhancement uses an exponential decay factor to adjust the position updates of explorers in order to optimize their efficiency when searching the solution space [25]. If \(X^{(i)}\) be the position of the \(j\)-th explorer at iteration \(i\), the update formula is as follows:  
\[X^{(i+1)}_{j}=X^{(i)}_{j}\times\left(2\times\exp\biggl{(}-\frac{4i}{\left( \xi\times Max_{i}\text{ter}\right)^{2}}\biggr{)}\right) \tag{20}\]  
where \(\xi\) is a random number within the interval [0, 1]. This improvement increases population diversity by combining a tent map and Cauchy mutation, thereby enhancing the algorithm's ability to avoid local optima and explore new areas [29; 30; 31]. Cauchy mutation is used when an individual's fitness value is less than the average fitness value of the population. Tent mapping is used when an individual's fitness value is greater than or equal to the average fitness value of the population. Cauchy mutation is determined.