## 4 NRBO Algorithm
### Newton-Raphson Search Rule (NRSR)  
The Newton-Raphson Method (NRM) employs Taylor series expansion to iteratively update the solution position, progressively refining it to identify the optimal solution. This approach enhances the exploration capability and accelerates convergence. Starting from a randomly selected initial solution, the subsequent solution position is updated along a specified direction. The position update process is as follows:  
\[x^{it+1}=x^{it}-\frac{f^{\prime}\big{(}x^{it}\big{)}}{f^{\prime\prime}\big{(}x ^{it}\big{)}} \tag{17}\]  
Here, \(x^{it}\) represents the current position of the solution. \(x^{it+1}\) represents the updated position of the solution and \(f(x)\) represents the fitness function.  
Based on the Taylor series expansion, the parameter \(\rho\) is introduced to guide the population in the correct evolutionary direction, thereby enhancing the efficiency of NRBO. Furthermore, improvements are incorporated based on the NRM proposed in [25; 26].  
\[\rho=a\cdot\Big{(}X_{\text{bet}}-X_{n}^{it}\Big{)}+b\cdot\Big{(}X_{r_{1}}^{it} -X_{r_{2}}^{it}\Big{)} \tag{18}\]  
\[\Delta X=rand(1,dim)\cdot\Big{|}X_{\text{bet}}-X^{it}\Big{|} \tag{19}\]  
\[\text{NRSR}=randn\cdot\frac{(Y_{\text{wor}}-Y_{\text{bet}})\cdot\Delta X}{2(Y _{\text{wor}}+Y_{\text{bet}}-2X^{it})} \tag{20}\]  
\[Y_{\text{wor}}=r_{3}\cdot(\text{Mean}(Z^{it+1}+X^{it})+r_{3}\Delta X) \tag{21}\]  
\[Y_{\text{bet}}=r_{3}\cdot(\text{Mean}(Z^{it+1}+X^{it})-r_{3}\Delta X) \tag{22}\]  
\[Z^{it+1}=X^{it}-randn\cdot\frac{(X_{\text{wor}}-X_{\text{bet}})\cdot\Delta X}{ 2(X_{\text{wor}}+X_{\text{bet}}-2X^{it})} \tag{23}\]  
Here, \(a\) and \(b\) are random numbers between 0 and 1. \(X_{n}^{it}\) represents the population \(n\) after the \(i\)-th iteration. NRSR is the NRM-based position update law. _randn_ is a standard normally distributed random number. \(X_{\text{bet}}\) and \(X_{\text{wor}}\) represent the better and worse populations in the vicinity, respectively. \(r_{1}\) and \(r_{2}\) are random integers between 0 and \(M\)\(Y_{\text{wor}}\) and \(Y_{\text{bet}}\) are two positions generated using \(Z^{it+1}\) and \(X^{it}\). \(r_{3}\) is a random number between 0 and 1. Thus, Equation (17) is updated as follows:  
\[X1_{n}^{it}=X_{n}^{it}-randn\cdot\frac{(Y_{\text{wor}}-Y_{\text{bet}})\cdot \Delta X_{n}}{2(Y_{\text{wor}}+Y_{\text{bet}}-2X_{n}^{it})}+\rho \tag{24}\]  
By replacing \(X_{n}^{it}\) with the better position \(X_{\text{bet}}\), the new position update formula is obtained:  
\[X2_{n}^{it}=X_{\text{bet}}-randn\cdot\frac{(Y_{\text{wor}}-Y_{\text{bet}}) \cdot\Delta X_{n}}{2(Y_{\text{wor}}+Y_{\text{bet}}-2X_{n}^{it})}+\rho \tag{25}\]  
The Equation (24) is effective for local search but has limitations in global exploration. In contrast, the Equation (25) emphasizes global search, albeit with limitations in local search. NRBO optimizes search performance by combining both approaches. Therefore, the new position for the next iteration is given by the following:  
\[X_{n}^{it+1}=r_{4}\cdot\left(r_{4}\cdot X1_{n}^{it}+(1-r_{4})\cdot X2_{n}^{it }\right)+(1-r_{4})\cdot X3_{n}^{it} \tag{26}\]  
\[X3_{n}^{it}=X_{n}^{it}-\delta\times\left(X2_{n}^{it}-X1_{n}^{it}\right) \tag{27}\]  
\[\delta=\left(1-\left(\frac{2K^{it}}{K_{\text{max}}^{it}}\right)\right)^{5} \tag{28}\]  
Here, \(K^{it}\) represents the number of iterations. \(K_{\text{max}}^{it}\) represents the maximum number of iterations, and \(r_{4}\) is a random number between 0 and 1.