## 3 Improved Golden Eagle Optimizer
### Algorithm Model
#### 3.1.2 Nonlinear Convex Decreasing Weight  
The nonlinear convex decreasing weight represents the degree of inheritance of the current location. The algorithm has strong global optimization capability (i.e., better exploration) when this value is large, because the population individuals inherit a large portion of the particle positions from the previous generation, and the algorithm has strong local optimization capability when this value is small, because the population individuals inherit little (i.e., better exploitation). To balance exploration and exploitation, a nonlinear convex decreasing weight is added in GEO. The greater inertia weight enables the golden eagle to cruise and explore more effectively in earlier phases of GEO. Golden eagles can attack prey more effectively because of their reduced inertial weight in later stages of GEO. Equation (8) shows that the location and moving step of the previous generation determine the golden eagle's updated position. In order for the previous person to affect the current individual, a nonlinear convex decreasing weight is added. Introducing the nonlinear convex declining weight is Equation (13).  
\[\omega(t)=\omega_{\text{ini}}(\omega_{\text{ini}}-\omega_{\text{end}})\bigg{(} 1-\bigg{(}\frac{t}{T}\bigg{)}^{m}\bigg{)}, \tag{13}\]  
where \(\omega_{\text{ini}}\) and \(\omega_{\text{end}}\) are the initial and final weight values when the algorithm has gone through a number of iterations equal to the minimum and maximum numbers of generations, [\(\omega_{\text{ini}}\)\(\omega_{\text{end}}\)] = [0.9 0.4]. \(t\) and \(T\) are the current and maximum generations. The different values of m represent the decreasing mode of inertial weight. Equation (14) represents the position update formula of the algorithm after introducing nonlinear convex decreasing weight.  
\[x_{i}^{t+1}=\omega(t)\cdot x_{i}^{t}+\Delta x_{i}, \tag{14}\]  
The value of m influences the weight's decreasing mode and further impacts the algorithm's performance during optimization, most notably its rate of convergence. Figure 3 depicts the nonlinear convex decreasing weight when m rises from 1 to 7. The following tests are conducted to investigate the value of m that is most advantageous to the algorithm. Six benchmark functions are chosen as experimental objects, 30 unique experiments are carried out, and seven different m values explain seven algorithms. The evaluation was conducted after generating the average convergence curve for these six benchmark test functions. The precise method involves finding the smallest number of iterations for each of the seven methods under the conditions of best convergence accuracy. The algorithm with the fewest iterations receives seven points, and so on, until the algorithm with the most iterations, which receives one point. The results of the seven algorithms for the six benchmark test functions are displayed in Table 1. Figure 3 and Table 1 show that the lowest score for m = 1, which reflects the linearly decreasing inertia weight, shows the algorithm's slowest convergence pace. In this paper, the value of m is chosen to be 4, because it has the highest score. It also exhibits the fastest convergence speed.