# Dynamic Heterogeneous Search-Mutation Structure-Based Equilibrium Optimizer
## 3 Dynamic Heterogeneous Search-Mutation Structure-Based Equilibrium Optimizer (DHSMEO)
### Dynamic Levy Mutation-Based Optimal Equilibrium Candidate-Refining Strategy
During the initial phase, a slowly decreasing control coefficient results in a high perturbation amplitude for refining the optimal equilibrium candidate, which raises the capability of improving global exploration and avoiding local optima; during the middle stage, a rapidly decreasing control coefficient makes the perturbation amplitude for refining the optimal equilibrium candidate diminish quickly, which contributes to maintaining the capability for escaping local optima and speeding up convergence, and during the final phase, a slow reduction in the control coefficient makes the perturbation amplitude for refining the optimal equilibrium candidate tend to be smaller, which contributes toenhancing the optimization accuracy and also ensures the capability for escaping local optima in the later stage to some extent. Let us sum up. DHSMEO improves the search capability for the population particles by integrating three improvement strategies from the above subsections to form a dynamic heterogeneous search-mutation structure. Among the three improvement strategies, the introduction of the dynamic dual-subpopulation adaptive grouping strategy improves population diversity and provides an effective information-exchange structure for the heterogeneous hybrid search strategy. The heterogeneous hybrid search-based concentration-updating strategy enhances the search efficiency. The dynamic Levy mutation-based optimal equilibrium candidate-refining strategy strengthens the capability of escaping local optima. Based on the above description, Figure 1 and Algorithm 1 present a flowchart and a pseudocode for DHSMEO. ```
1:Initialize the related parameters;
2:Randomly initialize the particle concentrations \(\mathbf{X}\) through Equation (1);
3:while\(l<l_{\text{max}}\)do
4: Perform fitness assessment for all particles;
5: Calculate the distance sequence \(\mathbf{d}\) of search particles through Equation (9);
6: Obtain the position sequence \(\mathbf{d}_{\text{ind}}\) of search particles by applying a sorting function sort(\(\cdot\)) to \(\mathbf{d}\) through Equation (10);
7: According to \(\mathbf{d}_{\text{ind}}\) and Equations (11)-(13), a kernel subpopulation \(\mathbf{X1}\) is constructed by assigning the last \(N_{\text{k}}\) search particles, and an auxiliary subpopulation \(\mathbf{X2}\) is constructed by assigning the remaining \(N_{\text{a}}\) search particles;
8:if\(l=0\)then
9: Decide the equilibrium candidates \(\mathbf{X}_{\text{eq(a)}}\), \(\mathbf{X}_{\text{eq($\beta$)}}\), \(\mathbf{X}_{\text{eq($\gamma$)}}\), \(\mathbf{X}_{\text{eq($\delta$)}}\) and \(\mathbf{X}_{\text{eq($\text{mean}$)}}\) according to the fitness values of search particles;
10:else
11: Decide the equilibrium candidates \(\mathbf{X}_{\text{eq($\alpha$)}}\), \(\mathbf{X}_{\text{eq($\beta$)}}\), \(\mathbf{X}_{\text{eq($\gamma$)}}\), \(\mathbf{X}_{\text{eq($\delta$)}}\) and \(\mathbf{X}_{\text{eq($\text{mean}$)}}\) according to the fitness values of search particles and refined optimal equilibrium candidate generated by the previous iteration;
12:endif
13: Construct the equilibrium pool \(\mathbf{X}_{\text{eq,pool}}\) through Equation (2);
14:for\(i=1:N_{\text{k}}\)do %X1
15:for\(j=1:D\)do
16: Update \(\mathbf{X1}_{(i,j)}\) through Equation (3);
17:endfor
18:endfor
19:for\(i=1:N_{\text{a}}\)do %X2
20:for\(j=1:D\)do
21: Update \(\mathbf{X2}_{(i,j)}\) through Equation (14);
22:endfor
23:endfor
24: Refine the optimal equilibrium candidate \(\mathbf{X}_{\text{eq($\alpha$)}}\) through Equations (20) and (22);
25:\(l=l+1\);
26:endwhile
27:return\(\mathbf{X}_{\text{eq($\gamma$0)}}\)
```  
**Algorithm 1** Pseudocode of DHSMEO