## 2 Materials and Methods
### Adaptive DBO for DPE
#### 2.3.2 Adaptive DBO Algorithm
The FDC measure ranges from \(-1\), indicating perfect anti-correlation, to 1, indicating perfect correlation. For minimization problems in DPE, large FDC values are considered misleading because the fitness tends to increase with distance from the global optimum. FDC values around zero are considered difficult, as they indicate very little correlation between fitness and distance from the global optimum. Low FDC values are seen as straightforward, in which fitness tends to increase as the global optimum is approached. Based on the FDC, we combine the spawning and foraging dung beetle populations and design a dynamic threshold \(p\) to determine the behavior selection, which is  
\[p=\frac{1+C_{f}}{2}. \tag{23}\]  
The adjustment of the populations for spawning and foraging dung beetles follows a heuristic principle. In each iteration, the algorithm assesses the required search capabilities and adjusts the number of dung beetles dedicated to both behaviors accordingly. When the fitness landscape is particularly challenging, the threshold \(p\) is modified to increase the number of foraging dung beetles, enhancing global exploration. Conversely, if the landscape demands finer local exploration, the number of spawning dung beetles is increased to strengthen local search capabilities. Additionally, when the FDC values are near zero, a balanced allocation of spawning and foraging dung beetles is maintained. With the same consideration, we aim to maintain the size of the search space when favoring global search, while narrowing it quickly when shifting towards local search. To achieve this, an adaptive convergence factor is designed as  
\[R=1-\frac{t}{T_{\text{max}}}+\frac{C_{f}}{2}\sin(\frac{\pi t}{T_{\text{max}}}). \tag{24}\]  
To demonstrate this visually, two satellite positioning scenarios are set up based on the simulation configurations described in Section 3. The first scenario involves eight visible satellites with fine geometric distribution, and the second scenario includes only four visible satellites with poor geometric distribution. These two cases represent relatively simple and complex optimization function modalities, respectively. As shown in Figure 3, the improved boundary convergence factor adjusts the search space according to the complexity of the problem posed to the population. The blue line corresponds to the linear decay of \(R\) in the original DBO algorithm. The two red curves represent the improved \(R\) values for the simple and complex modality scenarios, respectively. For the simple modality scenario, the factor decreases more slowly in the early stages, enhancing global search capability, and then decreases rapidly in the later stages to improve convergence speed. In contrast, for complex multi-modal problems, the convergence factor remains a high value throughout the search, ensuring that the global optimum is not missed. Hybrid Mutation Based on Ruggedness  
The original DBO algorithm faces challenges due to its limited convergence accuracy and a tendency to get trapped in local optima. Mutation operators are widely used to address these issues [32; 33]. The choice of operator depends on the problem modality: for single-peak problems, the Gaussian mutation operator is typically chosen to facilitate faster convergence to the global optimum, while for multi-peak problems, the Cauchy mutation operator is preferred as it allows for better exploration around local optima. However, varying capabilities are often desirable at different stages of the search process. The Cauchy-Gaussian mutation strategy, which employs a linear combination of Gaussian and Cauchy distributions, is a complementary approach designed to handle complex multi-peak scenarios [34]. The key challenge in hybrid mutation is determining the optimal coefficient that allows the mutation operator to best fit the current population. This optimal coefficient is dynamic and changes with the position of the population. In theory, such an optimal coefficient exists for each iteration.