## 4 Proposed HE-DEPSO Algorithm
### The Complete HE-DEPSO Algorithm  
The HE-DEPSO algorithm follows the usual structure of the DE algorithm in general, in which we have the stages of mutation, crossover, and selection. After generating a mutated vector \(V_{i,t}\) for the \(i\)-th individual of the population, as described in the previous section, we proceed to generate a test vector \(U_{i,t}\) for the individual \(X_{i,t}\) using the binomial crossover (Equation (23)). In order to reduce the sensitivity of the \(F\) and \(C_{r}\) parameter controls, the HE-DEPSO algorithm takes advantage of the individual-level adaptive parameter control of the SHADE algorithm to generate the parameter configuration of \(F\) and \(C_{r}\) for each individual. Finally, in the selection stage, the individuals that would make up the population for the next iteration are identified; this is done according to Equation (24). Based on the above description, the pseudocode of the complete HE-DEPSO is reported in Algorithm 1.  
Figure 2: Strategy selection curve; (**a**) Selection probability \(\alpha_{t}\), (**b**) Distribution of strategy selection.  
```
Input : Population size: \(NP\), Maximum number of iterations: \(t_{max}\), Dimensionality of the search space: \(D\) // Initialization
1 Iteration number: \(t=0\)
2 Randomly initialize the population: \(P_{0}=\{X_{t,0}\},i=1,2,...,NP\)
3 Fitness evaluation of initial population: \(\{f_{i,0}\}\)
4 Set all elements in \(\mathrm{M}_{\mathrm{F}}\) and \(\mathrm{M}_{\mathrm{C}}\) to \(0.8\) y \(0.5\) respectively
5 Set \(\mathrm{HE}=\varnothing\), \(\mathrm{HL}=\varnothing\)
6 Set \(p_{min}=0.1\), \(p_{max}=0.25\), \(k=0,c1=c2=2\)
7 Initialize best and best // Main loop
8 while\(t<t_{max}\)do
9\(S_{C}=\varnothing\); \(S_{F}=\varnothing\)
10 Update \(\mathrm{pb}^{\infty}\), \(a_{t}\) and \(\omega\)
11 Sort individuals in \(P_{t}\)
12 Generate \(\mathrm{E}_{t}\) and \(\mathrm{NE}_{t}\) from sets \(\{100 pb\%\}\) and \(\{NP-(100 pb\%)\}\)
13 Directly add \(\mathrm{E}_{t}\) to HE (Readjust size of HE if necessary)
14 Randomly select a pair of values from \(\mathrm{M}_{\mathrm{F}}\) and \(\mathrm{M}_{\mathrm{C}}\) to generate \(\mathrm{F}\) and \(\mathrm{C}r\) for all individuals
15 for\(i=1:NP\)do
16 Randomly select an index from \([1,NP]\) to obtain the associated \(F_{i,t}\) and \(Cr_{i,t}\) of \(X_{i,t}\)
17 Randomly select an individual \(X_{F_{i,t}}\) from \(\mathrm{E}_{t}\cup\mathrm{HE}\)
18 Randomly select an individual \(X_{F_{i,t}}\) from \(\mathrm{NE}_{t}\cup\mathrm{HL}\)
19 Generate a random number \(rand_{i}\) within \([0,1]\)
20 Generate the mutation vector \(V_{i,t}\) by: \(rand_{i}<a_{t}\), \(V_{i,t}=X_{i,t}+F_{i}\cdot(X_{F_{i,t}}-X_{i,t})+F_{i}\cdot(X_{F_{i,t}}-X_{i,t})\)
21 else
22\(V_{i,t}=\omega\cdot X_{i,t}+c_{1}\cdot r_{1}\cdot(X_{gbnet,t}-X_{i,t})+c_{2} \cdot r_{2}\cdot(X_{gbnet,t}-X_{i,t})\)
23 end if
24 Generate the trial vector \(U_{i,t}\) using binomial crossover and calculate its fitness
25 if\(f(U_{i,t})<f(X_{i,t})\)then
26\(X_{i,t+1}=U_{i,t}\)
27\(F_{i,t}\to S_{F_{i}}\); \(Cr_{i,t}\to S_{C}\)
28 else
29 Directly add \(X_{i,t}\) to HL (Readjust size of HL if necessary)
30\(X_{i,t+1}=X_{i,t}\)
31 end if
32 if\(f(U_{i,t})<f(X_{gbnet,t})\)then
33\(X_{gbnet,t}=U_{i,t}\)
34 end if
35 Update \(M_{\mathrm{C}}\) and \(M_{\mathrm{F}}\)
36\(k=k+1\) (if \(K>NP\to k=1\))
37\(t=t+1\)
38 end while Output :\(gbest\) and \(f(gbest)\)
```  
**Algorithm 1**The pseudocode of HE-DEPSO