## 6 Evaluated Algorithms
### FVIMDE Convergence Diagram  
The convergence curves for the CEC2022 benchmark Functions F1 through F6, as shown in Figure 2, illustrate how the FVIMDE optimizer performs across different types of optimization challenges. The curve for Function F1, a unimodal shifted and fully rotated Zakharov function, demonstrates a rapid initial decrease in the best value obtained. This behavior is typical for unimodal functions, where a direct approach can quickly lead to the global minimum. Function F2, a multimodal shifted and fully rotated Rosenbrock  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Function** & **Statistics** & **FVIDDE** & **FVIMMU** & **FVIMCR** & **FVIMSE** & **FVIMMUCR** \\ \hline F1 & Mean & 3.01E+02 & 1.14E+04 & 9.12E+02 & 1.10E+03 & 1.39E+03 \\  & Std & 5.  
function, shows a steep initial drop followed by a slower, more gradual descent. Moreover, Function F3, the shifted and fully rotated expanded Schaffer F6 Function, displays a more consistent and gradual descent, reflecting its multimodal landscape filled with numerous local optima that challenge FVIMDE's differentiation between local and global minima. Similarly, Function F4, the shifted and fully rotated non-continuous Rastrigin function, shows rapid early improvements that slows as the search continues, which is characteristic of a function landscape dotted with many local optima and the added complexity of non-continuity. Figure 2: Convergence diagram over CEC2022 benchmark functions (F1–F6). Function F5, the shifted and fully rotated Levy function, presents a very sharp initial improvement that plateaus--a pattern resulting from the function's long, narrow valleys, which require precise movements to reach the global minimum. Moreover, Function F6's curve, representing a hybrid function, shows several plateaus indicative of the diverse function landscape it comprises, testing both the exploitation and exploration capabilities of FVIMDE. The convergence curves for Functions F7 to F12, as shown in Figure 3, illustrate how FVIMDE performs across hybrid and composition functions. The curve for Function F7, a hybrid function, showcases a sharp initial descent and subsequent gradual improvements, suggesting that FVIMDE quickly locates a promising region but then switches to a more cautious approach to refine its search towards the global optimum. This indicates a balanced capability in exploring new areas and exploiting known good solutions. Figure 3: Convergence diagram over CEC2022 benchmark functions (F7–F12). Similarly, Function F8's convergence curve also begins with a steep drop but features fewer pronounced steps, which might point to a smoother landscape or more effective navigation strategies employed by FVIMDE. The relative flatness after the initial improvement might reflect fewer local optima or a more unified region drawing the optimizer toward the global optimum. For Function F9, a composition function, the curve displays a rapid initial decline followed by a prolonged, slow convergence, reflecting FVIMDE's ability to quickly identify viable paths to the optimum. However, the complexity of the composition, likely involving multiple overlapping functions, necessitates meticulous adjustments to fine-tune the solution. Function F10's curve, similar to F9, shows a steep initial descent that transitions into a steady, albeit slower, progression towards the minimum. This pattern underlines FVIMDE's proficiency in managing simultaneous optimization challenges presented by composition functions, effectively handling the layered landscapes. The convergence behavior for Function F11 mirrors that of Function F10 with a sharp initial drop. However, the extended plateau in the later stages of the iterations suggests a demanding optimization landscape where further improvements become increasingly difficult as FVIMDE approaches the vicinity of the global optimum. Moreover, the curve for Function F12 depicts a gradual but consistent decline throughout the iterations, suggesting continuous, incremental improvements.