## 5 Experimental Results and Analysis
### Comparison with DE, PSO, and Three State-of-the-Art DE Variants
#### 5.3.2 Convergence Properties  
The convergence properties can be summarized into four types, which are represented by the graphs presented in Figures 3 and 4.  
1. The convergence properties of the functions \(F_{1}-F_{4}\), \(F_{6}\), and \(F_{9}\) can be identified within the same class; this can be observed with the help of Figure 3a-d. In this type, the HE-DEPSO algorithm shows faster convergence than the other algorithms at \(D=10\), while at \(D=30\) it obtains a better mean error value.
2. In Figure 3e-h, the convergence curves of functions \(F_{5}\) and \(F_{8}\) are presented, which are similar to the corresponding ones for functions \(F_{10}\), \(F_{16}\), \(F_{17}\), and \(F_{20}\). All algorithms have different degrees of evolutionary stagnation or exhibit slow evolution for these functions.
3. The convergence curves of the functions \(F_{12}-F_{15}\), \(F_{18}-F_{19}\), \(F_{23}\), and \(F_{29}-F_{30}\) are similar to the curves of the functions \(F_{7}\) and \(F_{24}\) (see Figure 4a-d) obtained for the HE-DEPSO algorithm. In this type, HE-DEPSO presents a fast convergence at the beginning and, subsequently, it keeps evolving downward.
4. The evolutionary process presented in the convergence curves of the functions \(F_{22}\) and \(F_{27}\) follows the same trend as the convergence curves of the functions \(F_{25}\) and \(F_{28}\), as can be seen in Figure 4e-h. Here, most of the algorithms present stagnation in an accelerated way, although some of them manage to keep evolving downward.  
Figure 3: _Cont._  
The experiments performed above prove the superiority of the proposed HE-DEPSO algorithm in solving different optimization problems. The reasons why the HE-DEPSO algorithm obtains such superior performance can be summarized as follows:  
1. The adaptive hybrid mutation operator implemented in HE-DEPSO is based on the self-adaptive mutation strategy of the DEPSO algorithm, which has proven to be helpful in tackling several complex optimization problems. On the other hand, the collaborative work between the \"DE/current-to-EHE/1\" mutation strategy with historical information of the elite individuals and the PSO algorithm strategy generates a good balance between exploration and exploitation at different stages of the evolutionary process.
2. The self-adaptive control of the \(F\) and \(Cr\) parameters adopted from the SHADE algorithm allows the mitigation of parameter sensitivity. In this way, the crossover probability and the scale factor are dynamically adjusted during the evolutionary process at the level of each individual, which can make the proposed algorithm suitable for a wider variety of optimization problems.  
Figure 3: Convergence curves for the functions \(F_{1}\), \(F_{3}\), \(F_{5}\), and \(F_{8}\), with \(D=10\) and \(30\). The horizontal and vertical axis represent the iterations and the mean error values for the \(31\) independent repetitions.  
Figure 4: Convergence curves for the functions \(F_{1}\), \(F_{3}\), \(F_{5}\) and \(F_{8}\), with \(D=10\) and \(30\). The horizontal and vertical axis represent the iterations and the mean error values for the 31 independent repetitions.