## 3 Background
### Aquila Optimizer (AO)  
This section presents the basic formulation of Aquila Optimization (AO) algorithm [26]. The AO algorithm generally imitates Aquila's social behavior to catch its target. AO is a population-based optimization algorithm that is equivalent to other metaheuristic (MH) algorithms and generates a population \(X\) of \(N\) solutions. To carry out this technique, the following equation was employed:  
\[X_{ji}=LB_{i}+r_{1}\times(U_{i}-L_{i}),\ \,j=1,2,.....,N\,i=1,2,\ldots,D \tag{1}\]  
where \(U_{i}\) and \(L_{i}\) are the search domain's boundaries. \(r_{1}\) represents random numbers whose values are in the interval \([0,1]\) and a \(D\)-dimensional space.  
The following phase in the AO technique is for either exploring or exploiting until the best solution is identified. According to [26], exploration and exploitation can be accomplished with two approaches.  
Exploration uses the best solution \(X_{new}\) and the average of the solutions (\(X_{avg}\)), and its mathematical formulation is as follows:  
\[X_{j}(t+1)=X_{new}(t)\times\left(\frac{1-e}{E}\right)+(X_{avg}(t)-X_{new}(t)* rand), \tag{2}\]  
\[X_{avg}(t)=\frac{1}{N}\sum_{i=1}^{N}X(t),\,\forall i=1,2,\ldots,D \tag{3}\]The greatest number of epochs is denoted by \(E\). \(\left(\frac{1-e}{E}\right)\) is used to control the searching ability of AO in the exploration stage. The Levy flight \(L\) and \(X_{new}\) are used in the exploration stage to upgrade the solutions, as introduced in the following equation:  
\[X_{j}(t+1)=X_{new}(t)\times L+X_{rand}(t)+(y-x)*random, \tag{4}\]  
\[L=s\times\frac{r_{2}\times\sigma}{|r_{3}|^{\frac{1}{B}}},\,\sigma=\left(\frac {\Gamma(1+\beta)\times sine(\frac{\pi B}{2})}{\Gamma(\frac{1+\beta}{2})\times \beta\times 2^{(\frac{\beta-1}{2})}}\right),\,\beta=1.5,s=0.01 \tag{5}\]  
In Equation (5), \(r_{2}\) and \(r_{3}\) are values that are generated at random. \(X_{rand}\) refers to a random solution. Moreover, a spiral shape is formed by using the two parameters \(y\) and \(x\), which are defined as:  
\[x=R\times sin(\theta),\,y=R\times cos(\theta) \tag{6}\]  
\[R=r_{1}+Q\times D_{1},\,\theta=-S\times D_{1}+\theta_{1},\,\theta_{1}=\frac{3 \times\pi}{2},\,Q=0.00565,\,S=0.005 \tag{7}\]  
where \(r_{1}\in[0,20]\) is a random value.  
Similarly to exploration, the initial strategy employed in [26] to boost the solutions in the exploitation phase was based on \(X_{new}\) and \(X_{avg}\), and it is defined as:  
\[X_{j}(t+1)=(X_{new}(t)-X_{avg}(t))\times\alpha-r+(U\times r+L)\times\delta,\,U =(U-L) \tag{8}\]  
where \(\alpha\) and \(\delta\) are the adjustment parameters used during the exploitation phase. \(r\in[0,1]\) refers to a random value. Moreover, in the second strategy of exploitation, the solution will be updated by using the quality function \((QF)\), \(L\), and \(X_{new}\). This can be achieved with the following formula:  
\[X_{j}(t+1)=QF\times X_{new}(t)-G(G_{1}\times X(t)\times r)-G_{2}\times L+r \times G_{1} \tag{9}\]  
\[QF(t)=e^{\frac{2\times(t)-1}{(1-E)^{2}}} \tag{10}\]  
where \(G_{1}\) and \(G_{2}\) represent a parameter of the motions applied to track \(X_{b}\) and a parameter that decreases from 2 to 0, respectively; these parameters are defined as:  
\[G_{1}=2\times r()-1,\,G_{2}=2\times(1-\frac{e}{E}) \tag{11}\]  
Figure 1 depicts the steps of AO.  
Figure 1: Flowchart for AO.