## 3 Methodology of Solution
### Master Stage
To obtain a relation between function \(f(x)\) and its first and second derivatives, the [PERSON]'s series expansion was applied, considering its first two terms, which produced the relation defined in Equation (17):  
\[f(x)=f\Big{(}x_{j}^{t}\Big{)}+\Big{(}x-x_{j}^{t}\Big{)}f^{\prime}\Big{(}x_{j} ^{t}\Big{)}+\frac{\Big{(}x-x_{j}^{t}\Big{)}^{2}}{2}f^{\prime\prime}\Big{(}x_{ j}^{t}\Big{)}. \tag{17}\]  
Now, if we define \(\lambda\) as \(x_{j+1}^{t}-x_{j-1}^{t}\), and we also combine Equations (14)-(16), then the function values at points \(x_{j+1}^{t}\) and \(x_{j-1}^{t}\) can be obtained as presented in Equations (18) and (19):  
\[f\Big{(}x_{j+1}^{t}\Big{)}=f\Big{(}x_{j}^{t}\Big{)}+\lambda(1- \tau)f^{\prime}\Big{(}x_{j}^{t}\Big{)}+\frac{\lambda^{2}(1-\tau)^{2}}{2}f^{ \prime\prime}\Big{(}x_{j}^{t}\Big{)}. \tag{18}\] \[f\Big{(}x_{j-1}^{t}\Big{)}=f\Big{(}x_{j}^{t}\Big{)}-\lambda\tau f ^{\prime}\Big{(}x_{j}^{t}\Big{)}+\frac{\lambda^{2}\tau^{2}}{2}f^{\prime\prime }\Big{(}x_{j}^{t}\Big{)}. \tag{19}\]Simultaneously, solving Equations (18) and (19) for \(f^{\prime}\big{(}x_{i}^{t}\big{)}\) and \(f^{\prime\prime}\big{(}x_{i}^{t}\big{)}\) yields Equations (20) and (21):  
\[f^{\prime}\Big{(}x_{j}^{t}\Big{)}=\frac{\tau^{2}f\Big{(}x_{j+1}^{ t}\Big{)}+(1-2\tau)f\Big{(}x_{j}^{t}\Big{)}-(1-\tau)^{2}f\Big{(}x_{j-1}^{t} \Big{)}}{\tau(1-\tau)\Big{(}x_{j+1}^{t}-x_{j-1}^{t}\Big{)}}. \tag{20}\] \[f^{\prime\prime}\Big{(}x_{j}^{t}\Big{)}=\frac{2\tau f\Big{(}x_{j +1}^{t}\Big{)}-2f\Big{(}x_{j}^{t}\Big{)}+2(1-\tau)f\Big{(}x_{j-1}^{t}\Big{)}}{ \tau(1-\tau)\Big{(}x_{j+1}^{t}-x_{j-1}^{t}\Big{)}^{2}}. \tag{21}\]  
By combining Equations (20) and (21) in Equation (14), the evolution rule of the [PERSON] metaheuristic algorithm takes the form defined in Equation (22):  
\[x_{j}^{t+1}=x_{j}^{t}+\frac{\tau^{2}f\Big{(}x_{j+1}^{t}\Big{)}+(1-2\tau)f \Big{(}x_{j}^{t}\Big{)}-(1-\tau)^{2}f\Big{(}x_{j-1}^{t}\Big{)}}{2\tau f\Big{(} x_{j+1}^{t}\Big{)}-2f\Big{(}x_{j}^{t}\Big{)}+2(1-\tau)f\Big{(}x_{j-1}^{t} \Big{)}}\Big{(}x_{j-1}^{t}-x_{j+1}^{t}\Big{)}. \tag{22}\]  
Note that the evolution rule presented in (22) depends on the parameter \(\tau\) which was defined as recommended in [21] as presented in Equation (23):  
\[x_{j}^{t+1}=x_{j}^{t}-\frac{f^{\prime}\Big{(}x_{j}^{t}\Big{)}}{f^{\prime\prime }\Big{(}x_{j}^{t}\Big{)}}, \tag{14}\]0  
**Remark 1**. : _Based on the recommendations of the authors in [21], the evolution rule (22) can be improved by considering the position of the best optimal solution at the current iteration (i.e., \(x_{best}^{t}\)) which generates the evolution rule to explore and exploit the solution space that is defined in Equation (24)._  
\[x_{j}^{t+1}=x_{j}^{t}-\frac{f^{\prime}\Big{(}x_{j}^{t}\Big{)}}{f^{\prime\prime }\Big{(}x_{j}^{t}\Big{)}}, \tag{14}\]1  
_where \(r_{1}\) and \(r_{2}\) are random numbers between 0 and 1, \(t_{\max}\) is the maximum number of iterations, and \(\Gamma\) takes the form presented in Equation (25):_  
\[x_{j}^{t+1}=x_{j}^{t}-\frac{f^{\prime}\Big{(}x_{j}^{t}\Big{)}}{f^{\prime\prime }\Big{(}x_{j}^{t}\Big{)}}, \tag{14}\]2  
It is worth mentioning that the evaluation of the \(\Gamma\) function requires knowing the objective function values \(f\Big{(}x_{j-1}^{t}\Big{)}\), \(f\Big{(}x_{j}^{t}\Big{)}\), and \(f\Big{(}x_{j+1}^{t}\Big{)}\), respectively. The evaluation of these functions were obtained through the slave stage, which is presented below.