# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 3 MAOA Algorithm
### Modified Initialization Scheme of MAOA  
Similar to many other metaheuristic algorithms, the original AOA uses a random initialization procedure for the creation of the initial population. Although this approach is simple to implement, the stochastic nature of the conventional approach may not always guarantee good quality initial solutions because it does not fully exploit useful content from the search environment. Specifically, the random population initialization can lead to initial solutions being generated in local optima or in regions far from the global optimum, which compromises the robustness of the algorithm by increasing susceptibility to early convergence and reducing convergence efficiency. To address the shortcomings of traditional initialization, a modified initialization method is designed for the proposed MAOA, aiming to create an initial population with improved solution quality by leveraging the strengths of a chaotic map and a random oppositional-based learning mechanism (ROLM). Unlike the purely stochastic nature of random initialization, the ergodic behavior exhibited by the chaotic map allows for a more extensive search of the solution space, facilitating the creation of a more diverse initial population and thereby minimizing the likelihood of initialization in local optima. Additionally, the strong global search capabilities of ROLM are utilized to generate complementary opposite solutions for those initialized using the chaotic map, which can enhance the search process through broader navigation of the solution space. The mechanisms used in the modified initialization method of MAOA that engrave the chaotic maps and ROLM are described in detail below. The first half of the MAOA population, i.e., \(X_{n,d}\), for \(n=1,\ldots,N/2\) and \(d=1,\ldots,D\), is generated using the unique ergodic properties of a chaotic map to generate each dimensional component of the solutions in the first half of the population. Suppose that \(\xi_{0}\) denotes the initial value of the chaotic variable, which is randomly generated, and the updated value of this chaotic variable at any q-th iteration is denoted as \(\xi_{q}\), where \(q=1,\ldots,Q\), with Q being the maximum number of iterations for updating the chaotic variable. In this study, the circle chaotic map is employed in the modified initialization method to create the initial positions of the first half of the MAOA population. The circle chaotic map updates its chaotic variable at each q-th iteration as follows:  
\[\xi_{q+1}=\Big{|}\xi_{q}+b-\Big{(}\frac{a}{2\pi}\Big{)}sin\big{(}2\pi\xi_{q} \big{)},1\Big{|},\quad a=0.5,\quad b=0.2 \tag{11}\]  
Let \(\xi_{Q}\) denote the value of the chaotic variable after the final iteration (\(Q\)). Referring to the lower (\(X_{d}^{L}\)) and upper (\(X_{d}^{U}\)) boundary limits of the \(d\)-th dimension, the initial position in the \(d\)-th dimension for each solution of the first half of the population is generated as follows:  
\[X_{n,d}=X_{d}^{L}+\xi_{Q}\Big{(}X_{d}^{U}-X_{d}^{L}\Big{)},\quad\text{ for }n=1,\ldots,\frac{N}{2}\text{ and }d=1,\ldots,D \tag{12}\]  
The first half of the MAOA population is generated by iteratively applying Equations (11) and (12) across all dimensions (i.e., \(d=1,\ldots,D\)) for solutions that have a population index of \(n=1,\ldots,N/2\). Although the circle chaotic map helps reduce the likelihood of initial solutions being trapped in local optima, it does not completely prevent these initial solutions from being generated in regions that are far away from the global optimum, which may deteriorate the convergence characteristics of the algorithm. To address this issue, ROLM is adopted into the modified initialization method to create the second half of the MAOA population by producing complementary opposite solutions for the first half of the population created using the circle chaotic map.