## 6 Evaluated Algorithms
### Results on Benchmark of IEEE Congress on Evolutionary Computation 2022 (CEC2022)  
The FVIMDE algorithm showcases exceptional performance across the CEC2022 benchmark functions, as shown in Tables 4 and 5, significantly outperforming the compared optimizers. For instance, in functions such as F1 and F2, FVIMDE achieves the lowest mean values, highlighting its superior optimization ability. In F1, FVIMDE records a mean value of 3.01E+02, which is notably lower than that of FVIM (4.64E+03), STOA (1.88E+03), and other competitive algorithms like Chimp (2.42E+03) and CPO (1.58E+03). Similarly, in F2, FVIMDE leads with a mean value of 4.06E+02, surpassing other algorithms such as FLO (1.27E+03) and SOA(9.44E+02). When compared to other optimizers, FVIMDE consistently outperforms both classical algorithms like GA and SA, as well as modern metaheuristic optimizers like WOA and PSO. For example, in F7, FVIMDE achieves a mean value of 2.02E+03, outperforming SA (2.03E+03) and WOA (2.06E+03), which are commonly employed for solving complex optimization problems. Additionally, in F6, FVIMDE delivers a mean value of 1.80E+03, significantly lower than that of GA (1.53E+07) and SA (4.49E+03), further demonstrating its capacity to provide more accurate and consistent results. The robustness of FVIMDE is also evident in its lower standard deviations and standard errors across these functions, indicating not only its ability to find optimal solutions but also to do so reliably and consistently. This is a critical advantage in practical applications where stability and predictability are as crucial as achieving the best solution. As can be seen in Figures 12 and 13, the boxplot results of the FVIMDE algorithm compared to other optimizers across various functions in the CEC2022 benchmarks demonstrate that FVIMDE consistently exhibits lower error measures and less variability. In most F1-F12 functions, FVIMDE consistently ranks among the algorithms with the smallest error measures, indicating high accuracy and precision. In contrast, other optimizers, such as FLO and SSOA, show significantly higher error measures and greater variability, as evidenced by larger interquartile ranges and more frequent outliers. This suggests that FVIMDE offers robust performance across different test functions, maintaining stability where other algorithms struggle with either high error rates or inconsistency in their results. As seen in Tables 14 and 15, the Wilcoxon rank-sum test results provide a comprehensive analysis of how FVIMDE compares with various other optimizers across the CEC2022 benchmark functions (F1-F12). The test results are categorized as significant (S), non-significant (N), or equal (E), indicating whether FVIMDE's performance is significantly different, not significantly different, or statistically equal to that of the other algorithms. For most functions, FVIMDE demonstrates statistically significant superiority over other algorithms, with the majority of the results marked as S (significant). For instance, in functions like F1-F9, FVIMDE consistently achieves significant results againstall competitors. This is evidenced by extremely low p-values, such as 3.02E-11, indicating that the observed differences in performance are highly unlikely to be due to chance. These consistent S results across multiple functions underscore FVIMDE's strong optimization capabilities. However, there are a few instances where FVIMDE's performance is either equal or not significantly different from that of other algorithms. For example, in F4, while FVIMDE outperforms most optimizers, its comparison with the AO algorithm results in a non-significant (N) outcome, suggesting that both algorithms perform similarly for this particular function. Similarly, in F10, although FVIMDE shows a strong performance, the comparison with some optimizers results in a mix of S and N outcomes, reflecting closer competition.