## 2 Methods
### Improved Grey Wolf Optimizer (IGWO)
#### 2.2.1 The Integration of Cubic Chaotic Mapping and Reverse Learning Strategies  
4. Diversification of the Initial Population  
The positioning of initial solutions plays a crucial role in determining an algorithm's optimization efficacy and its ability to converge. Whereas traditional algorithms rely on pseudorandom number generators to position individuals across the solution space, this method often leads to an uneven distribution, causing overcrowding in certain regions. To mitigate this challenge, the introduction of chaos theory into the population initialization process marks a significant improvement. Its attributes of pseudorandomness, uniqueness, and thorough coverage ensure a more uniform distribution across the search space. The specific approach employs cubic chaotic mapping to delineate the initial population's positions, enhancing the algorithm's initial diversity and laying a robust foundation for the optimization journey. The formula utilized for this purpose is as follows:  
\[2y_{i+1}=4y_{i}^{3}-3y_{i} \tag{7}\]  
\[-1<y_{i}<1,\ y_{i}\
eq 0,\ i=0,1,\ldots,N \tag{8}\]  
\[x_{i}=X_{lb}+\frac{(X_{ub}-X_{lb})(y_{i}+1)}{2} \tag{9}\]where \(x_{i}\) denotes the mapped position of a grey wolf individual, and \(X_{lb}\) and \(X_{ub}\) delineate the solution space's lower and upper bounds, respectively. The process for initializing the grey wolf population through cubic chaotic mapping unfolds in several key steps: Firstly, commence by randomly generating a d-dimensional vector within the [\(-\)1,1] range, serving as the prototype individual. Then, apply Equation (7) to iterate over each dimension of the initial vector, thereby generating N \(-\) 1 subsequent individuals. Finally, utilize Equation (8) to transpose the values derived from cubic chaotic mapping into the coordinates of the grey wolf individuals.  
Elimization of the Initial Population Enhancing the exploratory range of the initial population's individuals and elevating the probability of identifying superior solutions necessitate the implementation of a lens imaging reverse learning technique. This method inverts the position of the current grey wolf individuals, yielding their lens imaging inverse solutions. If x\({}_{\text{j}}\) symbolizes the present grey wolf individual and \(x_{j}^{*}\) denotes its lens imaging inverse solution, the relationship can be articulated as follows:  
\[x_{j}^{*}=\frac{a_{j}+b_{j}}{2}+\frac{a_{j}+b_{j}}{2k}-\frac{x_{j}}{k} \tag{10}\]  
where \(a_{j}\) and \(b_{j}\) are the minimum and maximum values found within the j dimension of the current population. The variable \(k\) represents the scaling factor of the lens, which is predetermined to be 12,000.  
The initialization of the grey wolf population with the elite chaotic reverse learning strategy unfolds through a meticulously designed process:  
* Initial Population Creation Begin by randomly generating N initial grey wolf individuals \(\mathbf{X}=[\mathbf{x_{i1}},...,\mathbf{x_{id}}],\text{i}=1,...,\text{N}\), each denoting the ith wolf's position in the dth dimension.
* Chaotic and Reverse Transformations Apply Equation (7) to perform a chaotic transformation on population \(\mathbf{X}\), resulting in a chaotic population \(\mathbf{Y}\). Concurrently, implement Equation (9) to execute a lens imaging reverse transformation on population \(\mathbf{X}\), yielding the lens imaging reverse population \(\mathbf{Z}\).
* Elite Population Formation Evaluate and rank the grey wolf individuals within populations \(\mathbf{Y}\) and \(\mathbf{Z}\) based on their fitness values. Select the top N individuals from this evaluation to assemble the elite chaotic reverse population \(\mathbf{P}\).  
New Initial Population Selection: Sort the combined pool of individuals from population \(\mathbf{P}\) and the original population \(\mathbf{X}\) according to their fitness values. The foremost N individuals are then chosen to form the renewed initial grey wolf population.  
\[\mathbf{X}^{{}^{\prime}}=\left[\overset{\cdot}{x_{i1}},\dots,\overset{\cdot}{ x_{id}}\right] \tag{11}\]