## 2 Materials and Methods
### Adaptive DBO for DPE
#### 2.3.3 Convergence Performance Test  
The convergence performance of the proposed ADBO algorithm is evaluated using the benchmark test functions from the 2020 IEEE Congress on Evolutionary Computation (CEC2020) [36], which are a collection of synthetic mathematical optimization problems widely used to evaluate the search efficiency and robustness of optimization algorithms. The characteristics of the functions are summarized in Table 2. Due to the limited variants of the DBO algorithm, we compare it with seven other popular swarm intelligence algorithms: PSO, GWO, PIO, HHO, SSA, PO, and the basic DBO. To ensure the fairness of the experiment, the initial population size for all algorithms is set to 30, the maximum number of iterations to 500, and the search dimension to 30. To eliminate randomness, the evaluation criteria used are the mean and standard deviation of the solution results. Each algorithm is independently run 100 times on each test function. The parameters for the comparison algorithms are listed in Table 3, with ADBO and DBO sharing the same parameter settings. The experiments were conducted on a computer system with an Intel(R) i7-12700H CPU and 16 GB RAM running a 64-bit version of Microsoft Windows 11. The source code was implemented in the MATLAB (R2024a) programming environment. All subsequent experiments were performed on the same equipment. Table 4 presents the mean, standard deviation, and ranking of the solution results for ADBO and the comparison algorithms across each function in CEC2020. The ranking of the eight algorithms prioritizes the mean value. As shown in Table 4, ADBO achieves the top rank (i.e., rank 1) on seven out of ten benchmark functions, based on the mean value of each algorithm's results. It not only excels in unimodal and basic functions but also shows significant advantages in hybrid and composition functions. Notably, in \(F_{4}\), ADBO, HHO, and PO are effectively tied for first place, with minimal performance differences. In the first four functions, except for \(F_{1}\), where ADBO is not ranked first, it performs optimally in all other functions. ADBO secured the top ranking in all hybrid and composition functions, except for function \(F_{9}\), where PSO and SSA slightly outperformed it. These results demonstrate that ADBO consistently finds better solutions for these functions while maintaining high stability in its solutions. Overall, ADBO achieved the lowest total rank, indicating its superior performance compared with the other algorithms. In particular, ADBO not only offers significantly better solution accuracy than the original DBO algorithm but also exhibits lower variance. The effectiveness of these improvements is further validated by the Friedman mean rank test. As shown in Figure 4, ADBO consistently outperforms the other algorithms, thereby demonstrating the effectiveness of the improvements. The comparative convergence curves of the ADBO and seven other swarm intelligence algorithms are shown in Figure 5. These curves demonstrate that ADBO has a faster convergence rate than the other algorithms and can maintain a consistent search throughout the iterations, thereby reducing the likelihood of falling into local optima. The convergence curve of ADBO is also less volatile and stabilizes more effectively. Moreover, for most test functions, ADBO exhibits an accelerated convergence trend, indicating that its search capability enhances as iterations progress, allowing it to find better solutions more quickly.