## 2 Proposed Methodology
### Proposed Improved Reptle Search Algorithm (IRSA)  
The RSA is a newly developed optimization technique that can efficiently solve various optimization problems. However, while solving high dimensional nonconvex optimization problems, the RSA poses some drawbacks, such as slow convergence speed, high computational complexity, and local minima trapping [41; 42]. Therefore, to overcome these issues, some adjustments are proposed to the original RSA algorithm. Avoiding local minima trapping requires the solution candidates to explore the search space as widely as possible. Therefore, to enhance exploration, a sine operator was included in the high walking stage of the RSA algorithm. This adjustment was inspired by the dynamic exploration mechanism in the sine cosine algorithm (SCA) [43]. The sine operator provides a global exploration capability. Therefore, the inclusion of a sine operator in the IRSA can avoid local minima trapping by conducting a full-scale search of the solution space. Using the sine operator, in IRSA Equation (2) was replaced with the following equation. \[x_{jk}(\tau+1)=Best_{k}(\tau)+(r_{1}\times\sin(rand)\times|r_{2}\times Best_{k} (\tau)-x_{jk}|,\quad for\quad\tau\leq\frac{T}{3} \tag{11}\]  
where \(r_{1}\), \(r_{2}\), and \(rand\) are randomly chosen numbers between 0 and 1. \(x_{jk}\) is the current position, and \(Best_{k}\) is the best solution. The Levy flight [44] is a random process that follows the Levy distribution function. As suggested by yang [45]:  
\[levy=0.01\times\frac{u}{v^{\frac{1}{3}}} \tag{12}\]  
where \(u\) and \(v\) obey normal distribution. \[u\ \sim\ \left(0,\sigma_{u}^{2}\right),\ v\ \sim\ \left(0,\sigma_{v}^{2}\right) \tag{13}\]  
\[\sigma_{u}=\left(\frac{\delta(1+\zeta)sin\frac{\pi\zeta}{2}}{\delta\Big{[} \frac{1+\zeta}{2}\Big{]}\zeta*2^{\zeta-\frac{1}{2}}}\right)^{1/\zeta} \tag{14}\]  
\[\sigma_{v}=1 \tag{15}\]  
where \(\delta\) is the standard gamma function. \(\zeta\) is an important parameter in Levy fight that determines jump size. The lower value of \(\zeta\) results in small random steps. This enables the solution candidates to search the area nearest to the obtained solution, which improves exploitation capabilities. This improved exploitation guarantees global convergence. Therefore, instead of Equation (10) the Levy operator is used to update the position in the final stages of IRSA  
\[x_{jk}(\tau+1)=Best_{k}(\tau)+randn\times levy\oplus\Big{(}x_{jk}-Best_{k}( \tau)\Big{)},\ \text{for}\ \tau\leq T\ \text{and}\ \tau>3\frac{T}{4} \tag{16}\]  
where \(\oplus\) designates entry-wise multiplication, and \(randn\) is a uniformly distributed random number. The Levy random walk is represented in Figure 1.  
These improvisations greatly reduce the complexity of the algorithm. In the original RSA, Equations (4) and (5) are highly complex and increase the time complexity of the algorithm. However, with the above-mentioned adaptations, we can exclude these equations from the algorithm. This means that the proposed IRSA algorithms do not have to compute these equations, which results in an almost 3-to-4-fold reduction in time complexity. Improved global exploration, high efficiency, high convergence speed, and low time complexity are the improvements observed in the proposed technique.