## 4 Test Experiments and Discussion of DMMAEO
### Test Functions, Experimental Settings, and Performance Indicators  
The 58 test functions (TF1-TF58) primarily consist of two groups. The first group comprises 29 representative test functions (TF1-TF29), which can be classified as four types: unimodal (TF1-TF7), multimodal (TF8-TF13), fixed-dimension multimodal functions (TF14-TF23), and composite test functions (TF24-TF29). These 29 representative test functions are widely adopted by researchers [32; 33; 34]. Among the 29 representative test functions with four distinct categories, unimodal functions, characterized by a single global optimum, are adopted to evaluate the exploitation performance; multimodal functions, featuring multitudinous local optima, are employed in examining the exploration performance; fixed-dimension multimodal functions, possessing multitudinous local optima and fixed dimensions, serve as a benchmark for evaluating the algorithm's exploration capability within fixed-dimension search space; and composite functions, designed to mimic complex real-world optimization space with numerous local optima and diverse shapes in different areas, are utilized to test the balancing capability between exploration and exploitation. The second group comprises 29 CEC2017 test functions (TF30-TF58). The 29 CEC2017 test functions in the second group are more challenging to further evaluate the optimization performance of the algorithms than the 29 representative test functions in the first group. Therefore, a comprehensive analysis for the optimization performance of algorithms from various aspects can be achieved based on the 58 test functions. The detailed description for these 58 test functions can be obtained from [35]. Through the utilization of 58 test functions, two experimental designs are formulated in the test experiments to investigate the optimization capability of DMMAEO. The first experimental design investigates the impact of the three improvement mechanisms (dynamic multi-population guidance mechanism, dynamic Gaussian mutation-based sub-population concentration updating mechanism, and dynamic Cauchy mutation-based sub-population equilibrium candidate generation mechanism) on the canonical EO. The second experimental design involves analyzing the superiority of DMMAEO through comparing it to multiple recent promising algorithms, including GWO [5], HHO [6], EO [11], SMA [7], PSO [8], MPSO [36], MELGWO [37], LWMEO [20], and EBOwithCMAR [38]. In order to guarantee a fair comparison in the test experiments, identical experimental settings are applied to all the compared algorithms. The population size and particle dimension for each test function are taken as 30. The maximum iteration number for the test experiments is taken as 500. For achieving statistically meaningful results, each compared algorithm is independently implemented on each test function 30 times. All the test experiments are conducted by Matlab R2021b at the computer executing the 64-bit Microsoft Windows 10 operation system with Intel (R) Core (TM) i7-9750H 2.60 GHz CPU and 16 GB RAM. Based on the experimental settings and test functions mentioned above, the algorithm optimization ability is analyzed using four performance indicators: Ave., Std., Wilcoxon signed rank test, and Friedman test, which provide quantitative performance assessment from different aspects. Ave. represents the average fitness value derived from independent experiments conducted 30 times, which offers insight into the accuracy of the algorithm's optimization capability. Std. denotes the corresponding standard deviation derived from independent experiments conducted 30 times, which provides an assessment of the algorithm's stability. Wilcoxon signed rank test [39] is utilized in determining if the DMMAEO exhibits a significant superiority over other algorithms during the comparative experiments.