## 3 Proposed GLLCSA  
### Gradient Search Rule and Local Escaping Operator and Levy Flight Operator (GLL)  
In this part, we will introduce the GLL mechanism, which aims to achieve better performance for the CSA and prevent the CSA from sinking into the local optimum (LO). In this paper, the GSR, LEO, and LM operators are used to improve the personal position updating function of the CSA. The GSR strategy has shown great capability in many tasks [89; 90; 91; 92; 93]. In this study, the GSR strategy is mixed with the position update of the CSA to enhance the global search capability. The algorithm creates a population search method with strong robustness. The \(GSR\) can be expressed as:  
\[GSR=randn\times\rho_{1}\times\frac{2\Delta x\times x_{n}}{(x_{worst}-x_{best}+ \varepsilon)} \tag{3}\]  
where \(randn\) is a normally distributed random number, \(\Delta x\) means the value of the increment, and \(\varepsilon\) is a number from 0 to 0.1. \(x_{best}\) and \(x_{worst}\), respectively, represent the best and worst solutions gained in the search process. Equation (3) is better updated for the current solution. In addition, to better enhance the overall effect of the algorithm, \(\rho_{1}\) is then introduced into the equation to modify the GSP, as detailed below. The purpose of the optimization algorithm is to be able to search both locally as well as globally to achieve optimal results. Therefore, the GSR is updated by adaptive coefficients. Further, \(\rho_{1}\) is regarded as the vital variable in GBO to achieve this purpose, and it can be expressed as:  
\[\rho_{1}=2\times rand\times\alpha-\alpha \tag{4}\]  
\[\alpha=|\beta\times sin(\frac{3\pi}{2}+sin(\beta\times\frac{3\pi}{2}))| \tag{5}\]  
\[\beta=\beta_{min}+(\beta_{max}-\beta_{min})\times(1-(\frac{m}{M})^{3})^{2} \tag{6}\]  
where \(\beta_{min}\) and \(\beta_{max}\) are equal to 0.2 and 1.2, respectively; \(m\) expresses the number of iterations; \(M\) represents the blanket number of iterations. The GSR mechanism can be used to enhance the ability to search in the global space from the principle of the CSA. In Equation (3), \(\Delta x\) is determined by distinguishing the best solution (\(x_{best}\)) and a selected position (\(x_{r1}^{i}\)) (see Equations (7)-(9)) at random. Parameter \(\delta\) is defined by Equation (9). To improve exploration, a random number (\(rand\)) is mixed in Equation (9). \[\Delta x=rand(1:N)\times|step| \tag{7}\]  
\[step=\frac{(x_{best}-x_{r1}^{i})+\delta}{2} \tag{8}\]  
\[\delta=2\times rand\times(|\frac{x_{r1}^{i}+x_{r2}^{i}+x_{r3}^{i}+x_{r4}^{i}} {4}-x_{n}^{i}|) \tag{9}\]  
where \(rand(1:N)\) is a vector composed of \(N\) dimensions; \(r1,r2,r3\), and \(r4\) (\(r1\
eq r2\
eq r3\
eq r4\
eq n\)) are unequal integers chosen from 1 to \(N\) at random; \(step\) is the size of the step determined by \(x_{best}\) and \(x_{r1}^{i}\). On the other hand, the local escaping operator (LEO) is mixed in the CSA to improve the search performance in the local space. The principle is to transform the moving direction of the current solution into the direction toward the optimal solution so far into the local space, thus speeding up the algorithm convergence. Furthermore, \(LEO\) can be expressed in Equation (10):  
\[LEO=rand\times\rho_{2}\times(x_{best}-x_{n}) \tag{10}\]where \(rand\) is a random number from 0 to 1; \(\rho_{2}\) is based on a random number on \(\alpha\), which makes the vector have different step sizes. \(\rho_{2}\) is given by:  
\[\rho_{2}=2\times rand\times\alpha-\alpha \tag{11}\]  
Finally, Equations (12) and (13) aim to update the position of the current vector (\(x_{n}^{i}\)) established by the GSR and LEO mechanism.