## 2 Methodology
### Optimization Process  
The employed optimization process consists of a two-step implementation of an evolutionary algorithm framework, where Binary Differential Evolution (BDE) is employed to determine the activation states of individual antenna elements; then, Particle Swarm Optimization (PSO) is used to fine-tune the configuration of amplitude excitations. The hybrid approach leverages their strengths: MO-BDE excels in the exploration of different configurations, while MO-PSO offers fast convergence, ideal for fine-tuning amplitude weights in large-scale arrays. These algorithms are chosen for their proven effectiveness in addressing nonlinear, multi-objective optimization problems, particularly in array synthesis tasks [26; 27; 28; 29]. When combined with the dense packing and self-similar properties of Eisenstein fractal geometry, this approach enables precise control over critical performance parameters such as SLL and half-power beamwidth (HPBW), resulting in efficient and scalable antenna array configurations. The optimization process for achieving wideband performance is carried out using the lower and upper frequency boundaries (\(f_{l}\), \(f_{u}\)) as targets to ensure consistent radiation characteristics across the entire operational bandwidth. The operational lower frequency \(f_{l}\) is defined by the normalized spatial ratio \(d/\lambda=0.5\) and the upper frequency \(f_{u}\) is defined by \(d/\lambda=2\), where \(d\) is the spacing between elements of the array, \(f=c/\lambda\) is the frequency relation, and \(c\) is the constant of light propagation velocity [22]. As demonstrated in [13], optimizing the array at the upper frequency limit has a positive impact on performance at lower frequencies. Building upon this insight, the present work adopts a multi-objective optimization strategy aimed at enhancing the array response at the initial operating frequency while simultaneously preserving wideband behavior and minimizing the peak sidelobe level (SLL) [30]. The proposed optimization framework utilizes multi-objective functions to balance key design trade-offs, including element thinning, SLL suppression, and directivity enhancement in both frequency boundaries. Through the application of a mask function, it enforces constraints on the radiation pattern, encouraging a more uniform energy distribution across the angular spectrum. Lowering the peak SLL is critical for reducing interference and improving the overall quality of the radiation pattern [20; 29; 31; 32]. The objectives for thinning optimization in Algorithm 1 and excitation amplitude optimization in Algorithm 2 are  
\[OBJ_{1} =\frac{1}{n}\sum_{i=1}^{n}\left(AF_{f_{l}}(\theta_{i})-MASK_{f_{l}}( \theta_{i})\right)^{2} \tag{4}\] \[OBJ_{2} =\frac{1}{n}\sum_{i=1}^{n}\left(AF_{f_{u}}(\theta_{i})-MASK_{f_{u }}(\theta_{i})\right)^{2}\] (5) \[OBJ_{3} =1+\left[N_{T}-N\right]^{2} \tag{6}\]  
Equations (4) and (5) define the objective functions used to minimize beamforming error at two operating frequencies, \(f_{l}\) and \(f_{u}\), respectively. Each equation computes the mean squared deviation between the actual array factor \(AF_{f}\) and a reference mask \(MASK_{f}\) over \(n\) angular samples \(\theta_{i}\). \(MASK_{f}\) is derived from the main beam shape of the original \(AF_{f}\) response (no turned-off elements and no amplitude weights), where all sidelobe regions are replaced by a constant value corresponding to the target SLL. A target reduction of \(-10\) dB was applied across different stages and frequencies to define the desired envelope. The term \((AF_{f}(\theta_{i})-MASK_{f}(\theta_{i}))^{2}\) quantifies the squared error at each angular direction, and the summation accumulates this error across all directions, forming a least-mean-square-error (LMSE) criterion that ensures the radiation pattern conforms to the desired shape, suppressing SLL while maintaining main-lobe integrity. ```
1:Initialization: Generate an initial binary population representing active/inactive elements.