# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 3 MAOA Algorithm
### Construction of Unique Guidance for MAOA
In this case, a set variable \(\Psi_{N}=\{1,\ldots,N\}\) is constructed for the best-performing \(N\)-th solution, enabling it to select any other solution from \(\Psi_{N}\) as its unique guidance to explore wider regions of the solution space. Let \(\kappa_{N}\in\{1,\ldots,N\}\) represent the population index of any solution randomly selected from \(\Psi_{N}\), and let \(UG_{N}\) be the unique guidance assigned to the \(N\)-th solution. Then,  
\[\kappa_{N}=\text{Rand\_Select}(\Psi_{N}=\{1,\ldots,N\}) \tag{16}\]  
\[UG_{N}=X_{\kappa_{N}} \tag{17}\]  
The concept of unique guidance significantly enhances the resilience of MAOA to premature convergence by allowing each solution to perform a search under the influence of a distinct elite peer. In contrast to traditional global-best-based mechanisms, where all individuals are directed toward the same attractor, the unique guidance strategy promotes multi-attractor dynamics. Each solution is influenced by a superior individual selected from a personal set of better-performing peers, enabling diverse directional movement and preventing the entire population from collapsing into a narrow convergence cluster too early. This behavioral shift is particularly beneficial when navigating complex fitness landscapes that contain multiple deceptive local optima. If all solutions are attracted to a single best solution \(X^{best}\), and that solution resides in a suboptimal basin, the entire population may converge prematurely. However, under the unique guidance strategy, solutions are scattered across distinct regions, each influenced by different high-quality peers. This promotes competitive learning across niches, increases the diversity of search paths, and simultaneously maintains focus on high-potential regions. Additionally, even the best solution is exempt from rigid elitism, as it is guided by a randomly selected peer. This intentional disruption encourages continued exploration even among top individuals, helping to break the conventional greedy loop often observed in elitist metaheuristics. From a technical standpoint, the unique guidance mechanism transforms the guidance model from a static, globally centralized structure into a stochastic dominance-based framework. Let \(\Psi_{n}\) denote the set of solutions with better fitness than the \(n\)-th solution \(X_{n}\), such that \(|\Psi_{n}|=N-n\), where the population is sorted in descending order, i.e., from the worst to the best fitness values. Since the best solution \(X^{best}\) (at population index \(N\)) is always a member of \(\Psi_{n}\), the probability that any \(X_{n}\) selects \(X^{best}\) as its guide is as follows:\[P_{guide}(X_{n}\to X^{best})=\frac{1}{|\Psi_{n}|}=\frac{1}{N-n},\qquad\text{for $n<N$} \tag{18}\]  
This indicates that better-performing solutions (with larger indices \(n\)) have fewer options in \(\Psi_{n}\), and hence a higher probability of selecting \(X^{best}\) as their guide. On the other hand, the worse solutions (with smaller \(n\)) have access to a wider pool of elite guides, reducing the probability of following a single peer. This adaptive design reinforces exploitation among superior individuals while promoting exploration among weaker ones due to their access to greater guidance diversity. For the best solution \(X_{N}\), which has no superior individuals, guidance is randomly selected from all other population members:  
\[P_{guide}(X_{N}\to X_{k})=\frac{1}{N-1},\qquad\text{for $k\in\{1,2,\ldots,N-1\}$} \tag{19}\]  
This intentional randomness ensures that even the top solution participates in exploratory behavior, avoiding stagnation and reinforcing diversity at the elite level. Let \(D(t)\) denote the average pairwise Euclidean distance between solutions at iteration \(t\). In the original AOA, \(D(t)\to 0\) rapidly as all solutions converge to \(X^{best}\), leading to early stagnation. In contrast, the unique guidance strategy in MOAO preserves diversity such that \(D(t)>\epsilon\), where \(\epsilon\) is a positive constant representing a minimal diversity threshold.