# Dynamic Heterogeneous Search-Mutation Structure-Based Equilibrium Optimizer
## 4 Numerical Experiments and Discussion for DHSMEO
### Benchmark Functions, Experimental Configurations, and Performance Indicators  
The 39 typical benchmark functions (BF1-BF39) adopted in this section consist of four different types. As unimodal functions incorporating solely a single global optimum, BF1-BF7, are utilized to assess the exploitation performance. As multimodal functions incorporating numerous local optimal values, BF8-BF13 are commonly employed to gauge exploration performance. As fixed-dimension multimodal functions, BF14-BF23 feature multiple local optima and serve as benchmarks for evaluating exploration performance within a fixed-dimensional space. As composite functions, BF24-BF39 resemble the complexity of real-world optimization problems by incorporating multiple local optimal values and diversified landscape structures, providing a benchmark for evaluating exploration-exploitation trade-off capability. Among BF1-BF39, BF1-BF23 are derived from traditional benchmark functions widely used in this field [24; 25; 26; 27; 28], while BF24-BF39 originate from the CEC 2005 special session [29] and CEC 2017 special session [30]. The specifics for the aforementioned functions are presented within the corresponding research works [31]. Therefore, the employment of these 39 typical benchmark functions enables a comprehensive performance assessment across multiple aspects.  
Through employing the aforementioned 39 typical test functions, this section designs two experimental schemes to analyze the optimization performance of DHSMEO. The first experimental scheme aims to analyze the impact of the dynamic dual-subpopulation adaptive grouping strategy, the heterogeneous hybrid search-based concentration-updating strategy, and the dynamic Levy mutation-based optimal equilibrium candidate-refining strategy from the dynamic heterogeneous search-mutation structure introduced in Section 3 on the EO. The second experimental scheme is designed to analyze the superiority of DHSMEO by comparing DHSMEO with a collection of recently proposed algorithms, including EO [13], RSA [11], SMA [7], HGS [9], DMMAEO [21], DEEO [32], HRSA [33], ESMA [34], and AOAHGS [35]. The fairness of the two experimental schemes is ensured by implementing all algorithms under identical condition configurations. The population scale, dimension, and maximum iterations are assigned values of 30, 30, and 500, respectively. All compared algorithms are independently implemented 30 runs over test functions to obtain statistically significant results. Matlab R2021b serves as the experimental platform, running on a computer with a 64-bit Windows 10 operating system, an Intel i7-9750H CPU at 2.60 GHz, and 16 GB of RAM.  
With the aforementioned benchmark functions and experimental configurations, four performance indicators (average value (Avg), standard deviation (Std), Wilcoxon signed-rank test, and Friedman test) are adopted in evaluating optimization capability, which offers a comprehensive quantitative analysis from multiple perspectives. Avg reflects the average optimization capability across 30 independent operations, while Std measures the fluctuation range of the algorithm performance across the 30 independent runs. These two indicators serve to measure the algorithm's accuracy and stability, respectively. The Wilcoxon signed-rank test [36] is used to determine whether DHSMEO significantly outperforms other compared algorithms. A significance level of 0.05 is adopted, and the testresults are denoted as '\(1/0/-1\)', which indicates that DHSMEO shows superior, equal, or inferior optimization capability versus other algorithms. As a rank-based statistical analysis method, the Friedman test [37] is employed to assess the overall optimization capability from all compared algorithms. The results obtained from the Friedman test are presented as the Friedman mean rank and final rank, which provide an intuitive comparison of the overall performance and relative position of each algorithm in the experiments.