## 3 Dynamic Multi-Population Mutation Architecture-Based Equilibrium Optimizer (DMMAEO)
### Dynamic Cauchy Mutation-Based Sub-Population Equilibrium Candidate Generation Mechanism  
Within the dynamic multi-population guidance mechanism, the process of sub-population concentration updating is guided by sub-population equilibrium candidate. The sub-population equilibrium candidate is generated by selecting the optimal particle in the sub-population. To further enhance the exploration capability, dynamic Cauchy mutation is integrated into the process of sub-population equilibrium candidate generation. The Cauchy density function is attained by  
\[f_{\rm Cauchy}(z)=\frac{1}{\pi}\frac{\sigma_{2}}{(z-\mu_{2})^{2}+\sigma_{2}^ {2}}, \tag{17}\]  
where \(\mu_{2}\) and \(\sigma_{2}\) represent the location and scale parameters of Cauchy distribution, respectively. By setting \(\mu_{2}=0\) and \(\sigma_{2}=1\), this formula is simplified to generate random numbers in a way similar to the Gaussian mutation. On the basis of the random numbers generated by the Cauchy mutation, a dynamic Cauchy mutation-based sub-population equilibrium candidate generation mechanism is constructed by \[\mathbf{CP}_{\text{leq}(i)}(g)=\mathbf{P}_{\text{leq}(i)}(g)(1+\xi_{2,i}\mathbf{C}(z)), \tag{18}\] \[\xi_{2,i}=1-\frac{0.9g}{g_{\text{max}}}, \tag{19}\]  
where \(\mathbf{CP}_{\text{leq}(i)}(g)\) indicates the new sub-population equilibrium candidate created by dynamic Cauchy mutation; \(\mathbf{C}(z)\) denotes a step vector, which is composed of the random numbers generated through the Cauchy density function; and \(\xi_{2,i}\) represents a dynamic Cauchy mutation control coefficient. By employing the random numbers generated based on the Cauchy distribution and the dynamic Cauchy mutation control coefficient, the dynamic Cauchy mutation regulates the original equilibrium candidate to produce a new equilibrium candidate. In comparison to the Gaussian distribution used in the previous subsection, the Cauchy distribution exhibits the property of having a long flat tail, which can make the Cauchy mutation have a high probability of generating the new equilibrium candidate far from the original equilibrium candidate. This property can be used to strengthen the exploration ability through expanding the search space for the sub-population equilibrium candidates. Meanwhile, the dynamic Cauchy mutation control coefficient regulates the disturbance level of Cauchy mutation on the equilibrium candidate in a linearly decreasing manner. In this way, the exploration ability of the algorithm is greatly enhanced in the early stage, and the exploration ability of the algorithm is retained to some extent in the later stage. In a nutshell, the DMMAEO is developed for enhancing the optimization capability by introducing three enhancement mechanisms mentioned above. The utilization of dynamic multi-population guidance mechanism contributes to the increase in population diversity. The incorporation of dynamic Gaussian mutation during the sub-population concentration updating stage boosts the local search capability. Within the sub-population equilibrium candidate generation stage, the integration of dynamic Cauchy mutation serves to reinforce the global search ability. Based on the above introduction, Figure 1 presents the flowchart of DMMAEO, and the subsequent content outlines the steps of DMMAEO. **Step 1** Set the related parameters, including current iteration number \(g\), maximum iteration number \(g_{\text{max}}\), population size \(N\), particle dimension \(D\), sub-population number \(M\), and reorganization period \(R_{\text{p}}\). **Step 2** Randomly initialize the particle concentrations using Equation (1). **Step 3** Judge whether the iteration interval reaches the reorganization period \(R_{\text{p}}\). If so, dynamically create the multi-population structure using Equations (10), (11), and (13); otherwise, continue the process of sub-population concentration updating. **Step 4** Decide the sub-population equilibrium candidates \(\mathbf{P}_{\text{leq}(i)}(g)\).