## 3 Genetic Algorithm with Maximum-Minimum Crossover (GA-MMC)
### Optimization of Radiation Pattern Control of PAs
#### 3.1.2 Genetic Algorithms (GAs)  
GAs are quite useful in optimizing the design of PAs, as they optimize the calculation of excitation signals values of the array elements, which will be used for online control of the radiation pattern in several applications. In general, GA is a computational search method, which allows to converge to a set of optimal or quasi-optimal solutions through the application of [PERSON]'s Theory of Evolution and fundamentals of genetics [13, 14, 15, 16, 17, 18, 19]. In GAs, a population of possible solutions to the problem evolves according to probabilistic operators designed from biological metaphors, so there is a tendency that, on average, the individuals become increasingly better solutions as the evolutionary process goes on [16]. Therefore, GAs can be said as optimization methods for the solution of problems through \"survival of the fittest\" [60, 61, 62]. It has become a popular type of evolutionary computation, whose innovation is in a stochastic model that uses a population of solutions instead of a single one [63]. The evolutionary computing techniques are excellent for complex problems, for they are able to find better results in shorter periods of time [64]. In such problems, the optimization criterion involves choosing a set of variables to maximize (or minimize) a function of optimization (objective function), in order to explore search spaces of large dimensions, and manipulate a great number of requirements. In addition, GAs are applied to problems in all areas, because they are responsive to most problems, which may include: multiple optimization objectives; simple algorithms; capability of operating in discontinuous and non-differentiable search spaces; absence of a starting point to begin searching; faster processing and the possibility of being associated to other techniques. As for their operation, GAs operate concurrently with a set of points (population of individuals), which represent a search space of coded solutions [13]. Complex mathematical expressions are not used, such as in the case of the traditional gradient method [17]. This avoids using differential calculus or specific knowledge of the problem. Instead, much simpler rules and operators are applied over a fitness value, which are obtained by objective function for each individual in the population. Due to their versatility, GAs do not have a rigid model, but usually have a similar implementation flow [13, 14, 16]. This flow starts by generating a set of possible solutions to the problem, all coded according to a predetermined rule of representation. This set is nothing more than the initial population, which is modified for each full cycle of iteration, called generation. Individuals with higher fitness values have a greater chance to survive. They are also more likely to be selected as \"parents\" to generate a new individual [63, 65, 66]. In the flowchart of Figure 8, there is an example of GA, which is used in the GA-MMC [13]. The performance of a GA depends heavily on the parameters used (population size and rates of crossover and mutation), which are crucial to finding an optimal solution to the problem [65]. The choice of parameters may lead the evolution to a quick halt, with the population getting stagnant at a local optimum value, and thus does not meet the requirements for the problem. To avoid such premature convergence, it is common to increase the value of mutation genetic operator, in order to raise the genetic diversity of the population. On other hand, it can turn the algorithm in an almost random search. As for the optimum values, as with any optimization with multiple objectives, there is no single optimal solution, but a set of equally optimal solutions (multimodality). One of them should be chosen according to the requirements of application. Thus, to evaluate the optimization provided by the GA, one must compare the fittest individual with the intended objective.