# Dynamic Heterogeneous Search-Mutation Structure-Based Equilibrium Optimizer
## 4 Numerical Experiments and Discussion for DHSMEO
### Impacts of Improvement Strategies on EO
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Problem** & & **EO** & **DAEO** & **HUEO** & **LREO** & **DHSMEO** \\ \hline Unimodal & BF1 & Avg & \(4.572\times 10^{-41}\) & \(4.291\times 10^{-53}\) & \(0.000\times 10^{0}\) & \(1.364\times 10^{-16}\) & \(\mathbf{0.000\times 10^{0}}\) \\  & Std & \(4.015\times 10^{-41}\) & \(5.057\times 10^{-53}\) & \(0.000\times 10^{0}\) & \(0.000\times 10^{0}\) & \(0.000\times 10^{0}\) \\  & BF2 & Avg & \(9.844\times 10^{-24}\) & \(2.858\times 10^{-31}\) & \(0.000\times 10^{0}\) & \(3.752\times 10^{-87}\) & \(\mathbf{0.000\times 10^{0}}\) \\  & Std & \(9.957\times 10^{-24}\) & \(2.189\times 10^{-31}\) & \(0.000\times 10^{0}\) & \(1.559\times 10^{-86}\) & \(0.000\times 10^{0}\) \\  & BF3 & Avg & \(9.900\times 10^{-9}\) & \(1.390\times 10^{-14}\) & \(0.000\times 10^{0}\) & \(1.774\times 10^{-145}\) & \(\mathbf{0.000\times 10^{0}}\) \\  & Std & \(3.708\times 10^{-8}\) & \(1.969\times 10^{-14}\) & \(0.000\times 10^{0}\) & \(9.196\times 10^{-145}\) & \(0.000\times 10^{0}\) \\  & BF4 & Avg & \(2.767\times 10^{-10}\) & \(3.528\times 10^{-12}\) & \(0.000\times 10^{0}\) & \(1.781\times 10^{-76}\) & \(\mathbf{0.000\times 10^{0}}\) \\  & Std & \(2.546\times 10^{-10}\) & \(3.351\times 10^{-12}\) & \(0.000\times 10^{0}\) & \(9.686\times 10^{-76}\) & \(0.000\times 10^{0}\) \\  & BF5 & Avg & \(2.544\times 10^{1}\) & \(2.511\times 10^{1}\) & \(2.600\times 10^{1}\) & \(2.512\times 10^{1}\) & \(\mathbf{2.507\times 10^{1}}\) \\  & Std & \(1.59\times 10^{-and LREO in most cases. This result indicates that DHSMEO achieves the best performance through effectively integrating three improvement strategies to form a dynamic heterogeneous search-mutation structure. Table 9 further exhibits the corresponding results for Wilcoxon signed-rank test and Friedman test in the 39 typical benchmark functions. The Wilcoxon signed-rank test results, represented as '1/0/\(-\)1', denote that DHSMEO excels EO, DAEO, HUEO, and LREO across 32, 32, 28, and 31 benchmark functions. This result demonstrates that DHSMEO exhibits significantly better performance than the standard EO and other EO variants in most circumstances. The Friedman test results, presented in terms of [PERSON] mean rank and final rank in Table 9, assess the overall optimization performance of different EO variants. According to the Friedman test results, HUEO obtains the fourth place in the final rank with a [PERSON] mean rank of 3.462. This result shows that the heterogeneous hybrid search-based concentration-updating strategy improves the performance of EO by enhancing search efficiency. LREO achieves a [PERSON] mean rank of 2.872, placing it in third place in the final rank. This result indicates that the dynamic Levy mutation-based optimal equilibrium candidate-refining strategy is more effective than the heterogeneous hybrid search-based concentration-updating strategy in improving the optimization performance of EO by strengthening the capability of escaping local optima. The Friedman analysis assigns DAEO a [PERSON] mean rank of 2.769, ranking its placement at the second place of the final rank. This results show that the dynamic dual-subpopulation adaptive grouping strategy plays a key role in improving the performance of EO by boosting population diversity. The EO attains a [PERSON] mean rank of 4.436, situating it at the lowest position in the final rank. This result further confirms that the three enhancement strategies successfully increase the search capability for the standard EO. Meanwhile, with an outstanding Friedman mean rank of 1.462, DHSMEO secures the top position among the EO variants in the final rank. This result suggests that DHSMEO presents the best overall performance among the EO variants by combining these three improvement strategies simultaneously to form the dynamic heterogeneous search-mutation structure. Additionally, the convergence curves of EO, DAEO, HUEO, LREO, and DHSMEO on six benchmark functions are depicted in Figure 2 to visually assess their convergence capability.