# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 3 MAOA Algorithm
### Construction of Unique Guidance for MAOA  
Upon carefully examining the four arithmetic operators of the original AOA, as shown in Equations (8) and (10), it is evident that all these four operators share a common attribute, i.e., the current best solution (\(X^{best}\)) is consistently used to guide and adjust the search trajectory of all solutions throughout the optimization process. While this approach may help to accelerate the converging nature of the original AOA during the early stages of optimization, the frequency of updating \(X^{best}\) tends to decrease in the middle or latter stages due to a loss of population diversity. In unfavorable scenarios where \(X^{best}\) becomes trapped in a local optimum early in the optimization process, especially when dealing with complex optimization problems featuring challenging fitness landscapes, the remaining solutions are likely to be misguided toward these suboptimal regions, resulting in premature convergence. To handle the limitations encountered in the original AOA, the concept of unique guidance is introduced in the proposed MAOA. It is assumed that while \(X^{best}\) contains the most promising information at the current stage of optimization, it may not be ideal to use it as the sole guidance for all solutions due to the risk of premature convergence, particularly when dealing with complex optimization issues. Besides \(X^{best}\), other well-performing solutions in the population also contain valuable information that can be utilized to guide the search process. Compared to using only \(X^{best}\), leveraging multiple high-quality solutions to guide others is expected to direct the population toward promising regions while preserving population diversity. Motivated by this idea, the concept of unique guidance is incorporated into the proposed MAOA, enabling the search process of each solution to be influenced by a uniquely selected and well-performing guide. The mechanisms used to construct unique guidance for each solution in the MAOA population are detailed below. During the iterative search process, the MAOA population is first sorted based on the fitness values of the solutions in descending order, i.e., from the worst to the best fitness values. In other words, the worse-performing solutions are assigned smaller population indices, while the better-performing solutions are assigned larger population indices. After completing the sorting process, a set variable \(\mathbf{\Psi}_{n}=\{n,\ldots,N\}\) is defined for each n-th solution having a population index of \(n=1,\ldots,N-1\) to hold the population indices of other solutions that have better fitness values than the n-th solution. Evidently, the set \(\mathbf{\Psi}_{n}\)  
Figure 2: Flowchart of the modified initialization method for the proposed MAOA. associated with the worse-performing solutions contains more elements compared to those associated with better-performing solutions. Using the set \(\Psi_{n}\) for each n-th solution, a better-performing solution is randomly selected from \(\Psi_{n}\) to be used as unique guidance for adjusting the search trajectory of the n-th solution during optimization. Let \(\kappa_{n}\in\{n,\ldots,N\}\) represent the population index of a better-performing solution randomly selected from \(\Psi_{n}\), and let \(UG_{n}\) be the unique guidance assigned to the n-th solution. Then,  
\[\kappa_{n}=\text{Rand\_Select}(\Psi_{n}=\{n,\ldots,N\}) \tag{14}\]  
\[UG_{n}=X_{\kappa_{n}} \tag{15}\]  
where \(\text{Rand\_Select}(\cdot)\) is an operator used to randomly select an element from the given set \(\Psi_{n}\) for each n-th solution. It is noteworthy that Equations (14) and (15) are not applicable for the best-performing solution with the population index \(n=N\) because it has no other superior solution to serve as its unique guidance.