# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design
## 2 Antenna and Optimization Environment Description
### Optimization Environment  
The _Optimization Environment_ (OE) includes all the elements and the interactions between them that are required to properly conduct the optimization process. The definition of all the elements of the OE is fundamental to have a proper convergence of the optimization toward the function global optimum. The two main pillars of the environment are the optimization algorithm and the optimization problem (see Figure 4). The complete definition of the algorithm requires four choices: the first is which of the algorithms available in literature is the most suitable for the specific problem; then, it is necessary to select which are the optimization variables, how many objective function calls are required, and which set of algorithm internal parameters is the most suitable. For what concerns the optimization variables, they can differ from the problem design variables: in fact, it is possible to define the design variables as the combination of two or more optimization variables, or it is possible to reduce the problem size considering the physical knowledge or the constraints. The process of converting the optimization variables into the design variables consists of two steps: the first one is the decodification, i.e., the mapping procedure from the mathematical variables to the physical ones. The second step is the analysis of the variables with the feasibility function that takes into account the constraints on the problem variables. The most common constraints that are managed by the feasibility function are the box boundary conditions, i.e., the minimum and the maximum values that a variable can achieve. There are two main approaches to the feasibility function: it can edit the variables in order to satisfy the constraints, or it can reject the candidate solutions, bypassing the optimization problem. The feasible candidate solutions are used in the optimization problem definition to compute the system behavior and, thus, the performance parameters. Generally, engineering problems can have more than one performance index that should be considered in the optimization procedure. Once the performance parameters are computed, this information is fed back to the optimization algorithm by means of the cost function: this combines the performance indexes to obtain one (single-objective problem) or more (multi-objective problem) cost values. They can take into account different aspects, like the penalization of unfeasible solutions or solutions in which the system behavior is not acceptable. Once the optimization process is concluded, the final solution should be assessed. For the RA optimization problem, the general Optimization Environment can be tailored as follows. Figure 4: Optimization environment: it is composed by all the elements involved in the optimization process and it guarantees the quality of the final solution. The design variables of this problem are size of all the patches (\(24\times 24=576\) in this antenna) and, for each of the analyzed scan angles, the beam deviation factor. In order to reduce both the number of optimization variables and the computational effort in the system behavior calculation, it is possible to exploit the system symmetries: in fact, the required masks are symmetric in the \(\phi\) plane and the scanning requirements are symmetric for positive and negative \(\theta_{scan}\) angles. Due to this reason, the system geometry should have two symmetries, thus reducing the number of optimization variables related to the patch size to 144. Similarly, it is possible to compute the system behavior with only positive scan angles, and the number of analyzed \(\theta_{scan}\) is reduced to 4. The optimization variables are allowed to vary in the range \([0,1]\), while in the decodification process, these values should be mapped in the range \([0.5,4.5]\) for the patch size and \([-6,6]\) for the BDF. The only constraint directly connected to the design variables is the satisfaction of the box boundary conditions.