# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 3 MAOA Algorithm
### Modified Initialization Scheme of MAOA
The integration of ROLM in deriving the second half portion of the population as opposite solutions to the first half is expected to increase the likelihood of identifying more promising initial solutions by exploring a wider solution space. By applying ROLM to the first half of the population, the opposite solution for each member in the \(d\)-th dimension of the first half is calculated to generate the second half of the population, denoted as \(X_{N/2+n,d}\):  
\[X_{\frac{N}{2}+n,d}=X_{d}^{U}+X_{d}^{L}+r_{4}\times X_{n,d} \tag{13}\]  
where \(r_{4}\) is a random real-valued number lying between 0 and 1, and \(X_{N/2+n,d}\) represents the solution in the second half region of the population, calculated as the random opposite of the corresponding solution in the first half of the population. Similarly, the second half region of the MAOA population is generated by iteratively applying Equation (13) across all dimensions (i.e., \(d=1,\ldots,D\)) for solutions having a population index of \(n=N/2+1,\ldots,N\). The use of chaotic maps in the initialization phase provides a systematic yet pseudo-random means to ensure that solutions are well distributed across the search space. Unlike uniform random generation, which may result in clustered solutions or large unsampled regions, the circle chaotic map guarantees ergodicity, i.e., the ability to traverse every sub-region of the solution space given enough iterations. This makes the initial population structurally less prone to redundancy and more likely to contain individuals seeded in distinct basins of attraction. As a result, the likelihood of the entire population being concentrated near local minima is substantially reduced. Additionally, the incorporation of ROLM ensures that every solution generated by the chaotic map has a corresponding opposite solution, located across the search boundary in the solution space. This strategic mirroring mechanism enhances the algorithm's spatial reach from the outset. It not only boosts population diversity but also directly contributes to the global exploration potential of MAOA by effectively doubling the sampling range across each dimension. This broad initial coverage increases the probability of seeding individuals near the global optimum and simultaneously diminishes the collective bias toward local regions, especially in high-dimensional or multimodal optimization problems. From a theoretical standpoint, define \(P_{local}\) as the expected probability that a randomly generated solution falls into a local optimum's attraction basin. For conventional random initialization, this value depends on the landscape structure but remains relatively high in multimodal scenarios. The adoption of chaotic sequences modifies this expectation by reducing the clustering tendency of initial points. Due to the inherent non-repetitive and deterministic spread of the circle chaotic map, the distribution of points is better dispersed, which lowers \(P_{local}\) for a single solution, especially when compounded by dimensionality. Now consider the effect of ROLM. For every solution \(X_{n,d}\) in the \(d\)-th dimension initialized using the chaotic map, its oppositional pair \(X_{\frac{N}{2}+n,d}\) is generated according to Equation (13). This mirroring means that even if \(X_{n,d}\) lies within the vicinity of a local minimum, its opposite pair \(X_{\frac{N}{2}+n,d}\) likely resides outside this region, unless the problem space is symmetric and unimodal. Statistically, the inclusion of oppositional pairs increases the effective sampling coverage of the initialization process. This expansion of the initial population through both chaotic mapping and its ROLM-based opposites effectively broadens the search front. Since each ROLM-generated solution reflects across the boundaries of the decision space, the overall coverage of the initialization process is approximately doubled in terms of spatial reach compared to conventional random initialization.