## 3 Experimental Testing of Search Performance on IABCBAS Algorithms
### Convergence and Stability Analysis
Among the three high-dimensional multimodal functions, the WOA and SSA find the optimal values for \(f_{4}\) and \(f_{6}\), and the proposed IABCBAS algorithm maintains superior test performance; the proposed algorithm has much higher values for the \(f_{5}\) function than that of the other four algorithms, reflecting improved search capability. For the three low-dimensional multimodal functions \(f_{7}\) to \(f_{9}\), our algorithm reaches the optimal value under the solution, demonstrating better optimization performance under low-dimensional conditions. In terms of stability, only the ACO, SSA, and our algorithm find the optimal solution for \(f_{7}\), with their stability increasing in sequence. Our algorithm has the best stability for \(f_{8}\) and ranks second only to the WOA for \(f_{9}\). Across all algorithms, the accuracy on unimodal functions is higher than that on multimodal functions, and the accuracy on low-dimensional functions is higher than that on high-dimensional functions. The proposed IABCBAS algorithm achieves superior performance in both accuracy and stability. Regarding running time, different search mechanisms result in significant differences among algorithms. The average running times of the five algorithms are 3.994, 4.563, 5.597, 4.458, and 3.889 s, respectively. The proposed IABCBAS algorithm can generate multiple distinct solutions simultaneously, accelerating numerical comparisons and improving running time by 2.7% compared to the fastest PSO algorithm. The average convergence curves of the nine benchmark test functions are plotted to compare the convergence speed, accuracy, and ability to escape local optima among different algorithms, as shown in Figure 4. The vertical axis represents the fitness value, and the horizontal axis denotes the iteration number. Figure 4a-c demonstrate that the proposed IABCBAS algorithm exhibits faster convergence speed in unimodal functions, attributed to the chaotic initialization of the population. Simultaneously, the convergence accuracy of this algorithm significantly outperforms others. In both (a) and (c), the proposed algorithm escapes local extrema, whereas PSO, ACO, and WOA fail to do so. In the high-dimensional multimodal functions, ACO, WOA, SSA, and our algorithm achieve the optimal value of 0 in \(f_{4}\) and \(f_{6}\); thus, the convergence curves are not given due to early convergence. It is worth noting that our algorithm requires significantly fewer iterations than the other three algorithms and exhibits superior convergence performance. Figure 4d shows that the proposed IABCBAS algorithm falls into the local extreme value point in the 200 th\(\sim\)300 th generation, and there is a significant difference between the optimization value and the optimal value after 300 generations. However, the overall numerical curve amplitude is smaller and more stable than the other four algorithms. In the low-dimensional multimodal function, the numerical types of functions \(f_{8}\) and \(f_{9}\) are the same; only the former is shown in Figure 4f. All algorithms in Figure 4e,f converge to the optimal solution, but the proposed IABCBAS algorithm exhibits the fastest convergence speed. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Name** & **Algorithms** & \(f_{4}\) & \(f_{6}\) & \(f_{9}\) & \(f_{6}\) & \(f_{6}\) & \(f_{6}\) & \(f_{9}\) & \(f_{6}\) \\ \hline \multirow{4}{*}{\ \begin{tabular}{c} Optimal \\ value \\ \end{tabular}