## 4 Robust Learning-Based GWO Algorithm (RLGWO)
### Robust Tolerance-Based Adjust Searching Direction Mechanism (RTASDM)  
It is widely known that the grey wolf in the original GWO had a high possibility of getting stuck in local optima when operating in a large and complex search space. In Figure 1a, the hunting mechanism for a one-dimensional problem is presented, where the blue curve illustrates the object function. \(P_{\alpha}\) is the alpha position (the best solution) that leads the rest of the population. In Figure 1a, it is clear that each omega can move along the guiding direction of the alpha fitness value \(P_{\alpha}\) and the hunting group will be trapped in a local optimum after several iteration. Assume \(f(X_{\omega}^{n})^{k}\) indicates the objective value of the \(n\)th omega in the population at the \(k\)th iteration. \(X_{\omega}^{n}\) will be updated in the next iteration using Formula (13). Then, it will generate a new fitness denoted by \(f(X_{\omega}^{n})^{k+1}\). The total difference between \(f(X_{\omega}^{n})^{k+1}\) and \(f(X_{\omega}^{n})^{k}\) in the population can be described as shown in (14), where N is the population size. \[\sum_{n=1}^{N}\left(f(X_{\omega}^{n})^{k}\right)-\sum_{n=1}^{N}\left(f(X_{ \omega}^{n})^{k+1}\right)=A\ (A\in\mathbb{R}) \tag{14}\]  
Figure 1: The hunting mechanism of GWO in one-dimensional minimization. (**a**) \(P_{\alpha}\) falls into a local optimum; and (**b**) \(P_{\alpha}\) reaches the global optimum. After the course of iteration, the solution may finally converge to the optimum solution (global or local optimum). This situation indicates that \(A\) is more likely to be close to 0. Assuming that \(\varphi\) is a small value around 0, we can determine when the hunting group is going to converge by setting \(A\) that belongs to a range of \([-\varphi,\varphi]\). Therefore, Formula (14) can be rewritten as:  
\[\sum_{n=1}^{N}\left(f(X_{\omega}^{n})^{k}\right)-\sum_{n=1}^{N}\left(f(X_{ \omega}^{n})^{k+1}\right)\in[-\varphi,\varphi] \tag{15}\]  
To avoid the hunting groups getting stuck in the local optimum and to ensure the efficiency of the algorithm, the omega's search direction can be adjusted every time Equation (15) is satisfied. Nevertheless, when dealing with a large and complex space, we cannot rely on the circumstances described above to change the search direction of the omega. It can be clearly seen in Figure 1b that when the alpha position (global best solution) gets close to the global optimum, it leads the other members of the hunting group to search in the direction towards \(P_{\alpha}\). This also means that the solution of the generation \(k+1\) may not be improved by the alpha, while also satisfying the conditions shown in Equation (15). Therefore, depending on the potential ability of \(P_{\alpha}\), the hunting group can be led to a promising global optimum over the next few iterations [34]. After several iterations, as the number of satisfied conditions increases, meaning that there is no difference between the current solution and previous solutions, it can be concluded that the grey wolf is stuck around a local optimum. Therefore, the grey wolf needs to modify its search direction. Let \(h\) be a tolerance variable, where \(h\) is initially set to 0 and is used as a counter. In cases where (15) is satisfied, \(h\) can be updated via Equation (16), as follows. \[h=h+1 \tag{16}\]  
As the value of \(h\) becomes greater, the probability of becoming trapped in the local optima of a hunting group also increases [34]. However, in cases where \(P_{\alpha}\) is searching around the global optimum, as described above, the omega should not change their search direction, but rather move along the alpha direction. Therefore, we introduce the probability \(P_{adjust}\), which enables the omega to adjust its search direction. \(P_{adjust}\) can be achieved experimentally using Equation (17), below, where \(k\) is the current iteration and \(MaxIt\) is the maximum number of iterations.