# Adaptive Self-Scaling Brain-Storm Optimization via a Chaotic Search Mechanism
## 5 Experimental Studies
### Experimental Results and Analysis  
In this study, we compared the proposed algorithm with five other algorithms: BSO, CBSO, CGSA-P, DE and WOA. The parameters of these algorithms were set according to the optimal parameters provided in previous studies. To ensure the fairness of the comparison, we used the number of function evaluations rather than the number of iterations as the termination condition for each algorithm such that each algorithm paid the same calculation cost. The number of function evaluations was 5000, and the population size of all algorithms was 50. Thirty repeated experiments were performed for each algorithm on each test function, and then the results of 30 experiments were compared in terms of the _mean_ and _standarddeviation_. The results are shown in Table 1; the optimal and worst values are in bold and gray, respectively, and the second-best results in each benchmark function are underlined. The mean and standard deviation indicate the convergence performance of the algorithm and the dispersion degree or stability of 30 experiments compared with the current mean. Table 1 shows that ABSO-P obtained the largest number of optimal solutions of all the algorithms, indicating that it is very competitive compared with the other algorithms. ABSO-R showed the best performance for the non-complex functions F1 and F12 and obtained the second-best solution for F8, F25 and F28. ABSO had a remarkably better performance than BSO and CBSO, which implies that the proposed algorithm can help BSO jump out of local optima and fully employ its search ability. However, ABSO achieved similar results to CBSO for F10 and F11. In particular, for F11, the standard deviation of the ABSO results was only slightly higher than that of the CBSO results. In conclusion, the proposed improvement in BSO is effective. Compared with other parallel algorithms, DE showed a strong search performance, especially for F18, F19, and F20, for which it outperformed ABSO. However, on other benchmark test functions, ABSO had a better performance than DE, suggesting that the adaptive self-scaling method plays an effective role in complex search spaces and helps individuals jump out of local optima. Moreover, CGSA-P showed some advantages on specific functions, yet its search ability was poor when dealing with complex functions. In addition, both the original BSO algorithm and WOA failed to show a strong search performance. To analyze the experimental results more accurately, the Friedman test was used to obtain the average ranking of each algorithm on all test functions. In the nonparametric statistical test, the lower the ranking, the better performance of the algorithm. Table 2 shows the test results. ABSO-P ranked first (1.7931), while DE ranked second (2.9959); CGSA-P and ABSO-R showed similar searching abilities. Moreover, the Wilcoxon statistical test was used to verify the significance of the algorithm, and the results are summarized in Tables 3 and 4. In comparison with that of ABSO-P, the P value of each algorithm was lower than 0.05, indicating that ABSO-P has significant advantages over the other algorithms. In addition, the resistance values \(R^{+}\) and \(R^{-}\) reflected the excellent performance of ABSO-P.  
To more intuitively reveal the search performance and stability of the algorithm, we selected 4 representative functions (i.e., F3, F9, F12, and F26) from the 29 functions, and the corresponding convergence graphs and box diagrams are shown in Figure 2 and Figure 3, respectively. The ordinates are logarithmic to show the differences between these algorithms. As shown in Figure 2, when evaluated by the same function, ABSO-P converged rapidly to a good solution. Especially during the first half of iterations in F26, ABSO-P continued to converge downwards based on temporary convergence until the termination of iterations, which indicates that the adaptive self-scaling mechanism in ABSO improved the accuracy of the solution by effectively helping the algorithm jump out of local optima during the search process.