# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 3 MAOA Algorithm
### Additional Learning Phase of MAOA  
The unique guidance derived for each \(n\)-th solution in the previous subsection helps promote the population diversity of MAOA without compromising each solution's ability to explore promising parts of the solution space. To enhance this process further, mutual learning is introduced as an additional learning phase in the proposed MAOA, enabling each new solution to interact with other solutions that may contain valuable information in certain dimensions. For each \(n\)-th solution, after obtaining the new solution in each \(d\)-th dimension, the mutual learning phase allows the new solution (\(X_{n}^{New}\)) to learn from another randomly selected solution in the original population. The mechanisms of the mutual learning phase in the proposed MAOA are described as follows. Let \(\Phi_{n}=\{1,n-1,n+1,\ldots,N\}\) represent a set containing the population indices of other solutions that the \(n\)-th learner can interact with during the mutual learning phase. The new solution \(X_{n}^{New}\) can interact with all other solutions except itself. During the mutual learning phase, the \(n\)-th solution randomly selects an element from \(\Phi_{n}\) to determine the population index of the solution it will interact with when updating the \(d\)-th component of its new solution (\(X_{n,d}^{New}\)), as follows:  
\[\gamma_{n}=\text{Rand\_Select}(\Phi_{n}=\{1,n-1,n+1,\ldots,N\}) \tag{22}\]  
\[ML_{n,d}=X_{\gamma_{n},d} \tag{23}\]  
where \(ML_{n,d}\) represents the \(d\)-th component of the mutual learner that interacts with the n-th solution for dimension d (i.e., \(X_{n,d}^{New}\)). Notably, both Equations (22) and (23) are applied for \(D\) times to ensure that different dimensional components of a mutual learner can be contributed by different potential solutions from the original population. Let \(X_{n,d}^{Next}\) represents the \(d\)-th component of the updated solution for the \(n\)-th solution after interacting with the mutual learner (i.e., \(ML_{n,d}\)), calculated as follows:  
\[X_{n,d}^{Next}=X_{n,d}^{New}+r_{5}\Big{(}ML_{n,d}-X_{n,d}^{New}\Big{)} \tag{24}\]  
The updated solutions for all population members of MAOA are generated by iteratively applying Equations (22)-(24) across all dimensions (i.e., \(d=1,\ldots,D\)) for all solutions with population indices of \(n=1,\ldots,N\). Figure 4 illustrates the mutual learning mechanism incorporated into the proposed MAOA. The mutual learning phase serves as a crucial mechanism in MAOA to further sustain population diversity and mitigate stagnation risks, especially in the mid-to-late stages of the search process. While the unique guidance mechanism already promotes distributed learning by directing each individual toward a better peer, the additional learning phase introduces a second layer of interaction that facilitates decentralized knowledge diffusion. Figure 3: Flowchart for constructing unique guidance in MAOA. This form of structured interaction allows solutions to access complementary information from randomly selected peers, not necessarily limited to those superior in fitness. In doing so, MAOA avoids over-reliance on elite solutions and enables alternative paths to emerge, thus enhancing the algorithm's resilience against deceptive local optima. Most importantly, this mutual learning phase emulates a collaborative refinement strategy where partial solution components can evolve under the influence of different members of the population. Unlike conventional elitist-based updates that propagate dominant traits globally, this phase maintains diversity by decentralizing the influence map across the solution space. It effectively introduces a multi-source learning framework in which each dimension of a solution can be independently updated based on different references. This design fosters continual exploration and structural adaptability, which are crucial in complex or rotated landscapes where no single guide offers universal improvement.