# Ultra-Wideband Antenna Design for 5G NR Using the Bezier Search Differential Evolution Algorithm
## 2 Materials and Methods
### Bezier Search Differential Evolution Algorithm Description  
In the literature, many metaheuristic algorithms have been applied as optimizers in the domain of electromagnetics [2; 11; 12; 13; 14; 15; 16]. For example, these include the most popular Genetic Algorithm [17; 18] and Particle Swarm Optimization [19; 20] models, as well as the Gray Wolf Optimizer [21; 22], the Differential Evolution (DE) algorithm [23], the CMA-ES algorithm [24], Biogeography Based Optimization (BBO) [25; 26], and the Bald Eagle Search Algorithm [10]. The evolutionary algorithm used in this research paper to achieve the optimal antenna goal is the Bezier Search Differential Evolution Algorithm or BeSD algorithm [3]. BeSD is a stochastic evolutionary search algorithm based on matrix patterns and is a modified version of the Differential Evolution (DE) algorithm [23], which is a fast and reliable algorithm. It is designed as a global optimizer for real-valued numerical problems and was selected among a variety of candidate algorithms as it outperforms all of them when tested in benchmark functions, as it produces more accurate optimal results faster than the rest. As a general note, the ultimate advantage of the BeSD algorithm is its rapid convergence to the optimal solution, as already tested in research [27]. More specifically, the main advantages of the BeSD algorithm [3], which make it a better choice than other conventional optimization algorithms, are the following:  
* Its unique (Benzier polyonimials) mutation algorithm is multi-component, including a weighted-elitist component that strengthens exploitation. It differs from the mutation operators of conventional algorithms, providing an enhanced exploration aspect to BeSD. * In contrast to most conventional algorithms, its crossover process has no control parameters and is randomized, favoring both exploration and exploitation. * BeSD's characteristic structure showcases simplicity and efficiency, due to its limited computational complexity, which leads to accurate optimal solutions with a relatively quick convergence, compared to conventional algorithms. * Moreover, BeSD can be applied to parallel computing scenarios, as its patterns evolve separately, leading to a non-recursive algorithm. Briefly describing the optimization technique of the BeSD algorithm, the following definitions are considered. A random solution to the problem, consisting of as many variables as the size of the problem, is described as a pattern. A matrix of patterns is the total number of possible solutions used in the algorithm, which is reconstructed at the beginning of each iteration by randomly selecting N patterns from the pre-pattern matrix. Hence, the size of the pre-pattern matrix can be expressed as \(L\) times the size of the pattern matrix, M (Equation (2)). \[G_{p,j}\sim U(\mathrm{MinVar}_{j},\mathrm{MaxVar}_{j})\quad\text{where}\quad p =[1:L\cdot N],\;j=[1:D],\;1\leq L\leq 5 \tag{2}\]  
where \(MinVar\) and \(MaxVar\) are the lower and upper search boundaries of the \(j\)th variable of patterns, accordingly, and \(L\) is the size control value of the pre-pattern matrix \(G\). The values of the objective function, \(\mathcal{O}\mathcal{F}(. )\), for the matrix \(G\) are expressed by Equation (3). \[\mathrm{fit}G_{p}=\mathcal{O}\mathcal{F}(G_{p})\quad\text{where}\quad p=[1:L \cdot N] \tag{3}\]  
The initial value of the global solution, \(best_{global}\), and its objective function, \(min_{global}\), are given by Equation (4). \[[best_{global},min_{global}]=[G_{\delta},\text{fit}G_{\delta}]\quad|\quad\text{ fit}G_{\delta}=\text{min}(\text{fit}G),\ \delta\in\{1:L\cdot N\} \tag{4}\]  
Then, the iterative process described in Algorithm 1 takes place, until the predefined number of iterations is met, \(It=MaxIt\). A variety of index vectors are used in this algorithm, to balance both its exploration and exploitation properties. That is, \(v_{0},v_{1}\), and \(v_{2}\) are used to generate the initial pattern matrix \(M\) and \(fitM\) and the b-subjective vectors \(dw1\).