## 3 Dynamic Multi-Population Mutation Architecture-Based Equilibrium Optimizer (DMMAEO)
### Dynamic Multi-Population Guidance Mechanism  
Under the guidance of equilibrium candidates, particles in the canonical EO are regarded as a complete population for seeking the global optimum of the optimization problem. If the equilibrium candidates of EO are gathered together within the local optimum area, the whole population will dynamically evolve towards this region. This situation makes the whole population of EO unable to search the optimization space effectively. Hence, to design a novel population guidance mechanism for EO is pivotal, which contributes to enhancing population's search efficiency and population diversity in the optimization domain. For the achievement of this purpose, a dynamic multi-population guidance mechanism is introduced to overcome the limitations of EO's original guidance mechanism. The effectiveness of multi-population technology in searching the optimization space is mainly realized through processes including population division and information exchange among sub-populations. The traditional population division of multi-population technology is implemented based on a random sequence. Nonetheless, random sequence exhibits a remarkably prolonged period and poor correlation, which leads to the quality degradation of population division and the difficulty in realizing superior optimization performance. Drawing inspiration from [28], the dynamic multi-population guidance mechanism adopts a chaotic sequence for the process of population division. In this way, the whole population are divided through producing a set of strongly correlated chaotic sequences, which can enhance the population division quality and population optimization ability. The population division using a Logistic chaotic sequence is shown as. \[\mathbf{CS}(i+1)=\lambda\mathbf{CS}(i)(1-\mathbf{CS}(i)),i=1,2,3,\ldots,N-1, \tag{10}\] \[[\sim,\mathbf{I}_{\text{d}}]=\text{sort}(\mathbf{CS}), \tag{11}\]  
where \(\mathbf{CS}\) is a chaotic sequence generated by the Logistic map, whose number of elements is identical with the population size \(N\); the initial value \(\mathbf{CS}(1)\) is randomly selected between 0 and 1, and its value needs to be different from 0.25, 0.5, and 0.75; \(\lambda\) indicates a control parameter, which is set to 4 to make the Logistic map appear a state of complete chaos; sort\((\cdot)\) is a sort function, which is used to convert the chaotic sequence to a positional sequence; and \(\mathbf{I}_{\text{d}}\) represents the positional sequence. With Equations (10) and (11), the specific process of population division with \(M\) sub-populations in the dynamic multi-population guidance mechanism is implemented in the following way. Firstly, a chaotic sequence \(\mathbf{CS}\) is generated by Equation (10), whose length is consistent with the population size \(N\). Then, a positional sequence \(\mathbf{I}_{\text{d}}\) is obtained by Equation (11). Finally, according to the number of positional sequence \(\mathbf{I}_{\text{d}}\), the population particles are equally divided into \(M\) sub-populations to generate the multiple sub-population structure. With this generated structure, sub-population particles are updated under the guidance of the optimal equilibrium candidate in the sub-population instead of the equilibrium candidates in the whole population, which is shown as  
\[\mathbf{P}_{i}(g+1)=\mathbf{P}_{\text{leq}(i)}(g)+\Big{(}\mathbf{P}_{i}(g)-\mathbf{P}_{\text{leq }(i)}(g)\Big{)}\mathbf{H}_{i}(g)+(1-\mathbf{H}_{i}(g))\frac{\mathbf{F}_{i}(g)}{\mathbf{\xi}_{ i}(g)V}, \tag{12}\]  
where \(\mathbf{P}_{\text{leq}(i)}(g)\) represents the sub-population equilibrium candidate, which is the optimal equilibrium candidate in the local sub-population structure of \(i\)-th particle at iteration \(l\). During the process of sub-population updating, reorganization period \(R_{\text{p}}\) is applied to guide information communication across sub-populations. The value of \(R_{\text{p}}\) is crucial in the result of this information communication mode.