## 2 Methodology
### Proposed GSLEO-AEFA
#### 2.2.2 Local Escaping Operator (LEO)  
To improve the search capabilities of the gradient-based optimizer (GBO), local escaping operator (LEO) was initially developed by [PERSON] et al. [46] by adjusting the locations of the individuals in relation to a certain criteria. LEO improves the quality of solutions. In particular, it enhances the convergence of the algorithm and keeps it from becoming stuck in local optima. It is worth to note here that local escaping operator (LEO) from the gradient-based optimizer (GBO) has been utilized to enhance the local escaping functionality of GSLEO-AEFA without altering the LEO itself. By incorporating the LEO within the algorithm, we leverage its ability to escape local optima and converge towards the global optimum, thereby improving the overall performance of the GSLEO-AEFA.  
The best individual's position \(P_{\text{best}}\), two randomly created individuals' positions \(P_{r1}^{m}\) and \(P_{r2}^{m}\), two randomly selected positions \(P_{n}^{1m}\) and \(P_{2n}^{2m}\), and a new randomly generated solution \(P_{k}^{m}\) are used to generate a new solution \(P_{\text{LEO}}\), which has great performance [15]. LEO formulas are expressed in Equations (22) and (24). If \(rand<0.5\), Equation (22) is computed, otherwise Equation (24).  
\[P_{\text{LEO}}\ =P_{n}^{m}+f_{1}(u_{1}P_{\text{best}}\ -u_{2}P_{k}^{m})+f_{2} \rho_{1}(u_{3}(P_{n}^{2m}-P_{n}^{1m}))+u_{2}(P_{r1}^{m}-P_{r2}^{m})/2 \tag{22}\]  
\[P_{n}^{m+1}=P_{LEO} \tag{23}\]\[P_{\text{LEO}}\,=P_{\text{best}}\,+f_{1}(u_{1}P_{\text{best}}\,-u_{2}P_{k}^{m})+f_{ 2}\rho_{1}(u_{3}(P2_{n}^{m}-P1_{n}^{m}))+u_{2}(P_{r1}^{m}-P_{r2}^{m})/2 \tag{24}\]  
\[P_{n}^{m+1}=P_{LEO} \tag{25}\]  
where \(P_{n}^{m}\) is the position being modified by LEO, \(rand\) is a random number within the range of [0, 1]. Within the range of [\(-\)1, 1], \(f_{1}\) and \(f_{2}\) are random numbers uniformly distributed.  
\[P1_{n}^{m},\ P2_{n}^{m}=lb+\text{rand}(\text{dim})\times(ub-lb) \tag{26}\]  
\[u_{1}=L_{1}\times 2\times rand+(1-L_{1}) \tag{27}\]  
\[u_{2}=L_{1}\times rand+(1-L_{1}) \tag{28}\]  
\[u_{3}=L_{1}\times rand+(1-L_{1}) \tag{29}\]  
where \(ub\) and \(lb\) denote the upper and lower boundaries, respectively, dim represents the solution dimension, \(n\) denotes the number of individuals in the population, \(m\) represents a decision variable in a given solution. \(u_{1}\), \(u_{2}\), and \(u_{3}\) are randomly generated variables. If \(k_{1}<0.5\), \(L_{1}\) set to 1, otherwise 0. \(k_{1}\) is number within the range [0, 1].  
\(\rho_{1}\) in Equations (22) and (23) is expressed as follows,  
\[\rho_{1}=2\times\text{rand}\times\alpha-\alpha \tag{30}\]  
\[\alpha=\left|\beta\times\sin\biggl{(}\frac{3\pi}{2}+\sin\biggl{(}\beta\times \frac{3\pi}{2}\biggr{)}\biggr{)}\right| \tag{31}\]  
\[\beta=\beta_{min}+(\beta_{max}-\beta_{min})\times\biggl{(}1-\biggl{(}\frac{i}{ i_{max}}\biggr{)}^{3}\biggr{)}^{2} \tag{32}\]  
where \(\beta_{min}\) and \(\beta_{max}\) are set to 0.2 and 1.2, \(i_{max}\) and \(i\) denote the maximum iteration and current iteration, respectively. To balance exploitation and exploration \(\rho_{1}\) is regulated by \(\alpha\).  
\(P_{k}^{m}\) in Equations (22) and (23) is computed as follows:  
\[P_{k}^{m}=\begin{cases}P