# 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  
Under the dynamic dual-subpopulation adaptive grouping strategy, the concentration-updating processes of the kernel subpopulation and auxiliary subpopulation with a heterogeneous hybrid search strategy guided by the equilibrium candidates and the optimal equilibrium candidate, respectively. Since the generation of the equilibrium candidates is influenced by the optimal equilibrium candidate, this makes the optimal equilibrium candidate become the central hub for algorithm guidance strategy. Therefore, a dynamic Levy mutation-based optimal equilibrium candidate-refining strategy is proposed in guiding the heterogeneous hybrid search-based concentration-updating process under the dynamic dual-subpopulation structure, which can ulteriorly strengthen the capability for escaping local optima. Levy flight characterizes the Levy distribution through random steps. According to the Mantegna method [23], a random step size for Levy flight is denoted as  
\[\mathbf{L}_{\text{f}}=\frac{\mathbf{\mu}}{|\mathbf{v}|^{\frac{1}{2}}}, \tag{17}\] \[\mathbf{\mu}{\sim}N(0,\sigma_{\mu}^{2}),\mathbf{v}{\sim}N(0,\sigma_{v}^{ 2}),\] (18) \[\sigma_{\mu}=\left\{\frac{\Gamma(1+\delta)\text{sin}\left(\frac{ \pi\delta}{2}\right)}{\Gamma\left(\frac{1+\delta}{2}\right)\delta\cdot 2^{ \frac{\delta-1}{2}}}\right\}^{\frac{1}{2}},\quad\sigma_{v}=1, \tag{19}\]  
where the following applies: \(\mathbf{L}_{\text{f}}\) is the Levy random step vector; \(\delta\) is assigned a value of 1.5; \(\Gamma(\cdot)\) is the Gamma function; and \(\mathbf{\mu}\) and \(\mathbf{\
u}\) represent two vectors obeying the Gaussian distribution. The step size of Levy flight is always small, but sometimes there will be a big jump. These characteristics can offer a deeper search mode for a more effective global search. Therefore, based on the obtained Levy random step length vector, a dynamic Levy mutation-based optimal equilibrium candidate-refining strategy is established in strengthening the capability for escaping local optima, which is expressed as follows:  
\[\mathbf{X}_{\text{eq(a)}}^{\prime}(l)=\mathbf{X}_{\text{eq(a)}}(l)(1+ \xi_{\text{L}}(l)\mathbf{L}_{\text{f}}), \tag{20}\] \[\xi_{\text{L}}(l)=0.9/(1+\exp(10\frac{l}{l_{\text{max}}}-5))+0.1,\] (21) \[\mathbf{X}_{\text{eq(a)}}(l)=\left\{ \begin{array}{ll}\mathbf{X}_{\text {eq(a)}}^{\prime}(l),&f\Big{(}\mathbf{X}_{\text{eq(a)}}^{\prime}(l)\Big{)}<f\Big{(} \mathbf{X}_{\text{eq(a)}}(l)\Big{)},\\ \mathbf{X}_{\text{eq(a)}}(l),&f\Big{(}\mathbf{X}_{\text{eq(a)}}^{\prime}(l)\Big{)} \geq f\Big{(}\mathbf{X}_{\text{eq(a)}}(l)\Big{)},\end{array} \right. \tag{22}\]  
where the following applies: \(\mathbf{X}_{\text{eq(a)}}^{\prime}(l)\) represents the optimal equilibrium candidate generated based on the dynamic Levy mutation; \(\xi_{\text{L(i)}}\) denotes the dynamic Levy mutation control coefficient; and \(f(\mathbf{X}_{\text{eq(a)}}^{\prime}(l))\) and \(f(\mathbf{X}_{\text{eq(a)}}(l))\) represent the fitness values of \(\mathbf{X}_{\text{eq(a)}}^{\prime}(l)\) and \(\mathbf{X}_{\text{eq(a)}}(l)\), respectively, which are employed for choosing the better optimal equilibrium candidate as the refined optimal equilibrium candidate. Equations (20)-(22) provide an optimal equilibrium candidate-refining strategy with a nonlinearly decreasing perturbation amplitude.