# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design
## 3 Evolutionary Algorithms
### Social Network Optimization  
Social Network Optimization (SNO) is a population-based algorithm that mimics the information sharing process in common online social networks. The population of this algorithm consists in the social network users that share their ideas and interact online.  
Each user is characterized by its opinion that it is shared by means of a post (out of the metaphor, the candidate solution of the optimization problem). The post is evaluated by the social network and it receives a visibility value (the cost value of the problem) that indicates how much it is probable that another user can read it [35].  
The online interaction takes place through two different networks: the _friend_ one, characterized by strong connections among users and by a slow evolution rate, and the _trust network_, characterized by weaker interactions and by an evolution based on the posts' visibility value. Each user exchanges opinions with other individuals and is influenced by both the people networks. The interaction is based on the following equation:  
\[\mathbf{o}(t+1)=\mathbf{o}(t)+\alpha[\mathbf{o}(t)-\mathbf{o}(t-1)]+\beta[ \mathbf{a}(t)-\mathbf{o}(t)] \tag{19}\]  
where \  
Another choice is to eliminate the solutions that go outside the search domain; then a new random solution is created in the domain. In this case the exploration is drastically increased, even if the convergence can be worsened for the algorithms that work with trajectories, like, for example, the Particle Swarm Optimization. In fact, this boundary condition completely destroys the original trajectory. Its mathematical formulation is [36]:  
\[\hat{x}_{i}=\left\{\begin{array}{ll}r,&x_{i}<L_{i}\\ r,&x_{i}>U_{i}\\ x_{i},&\text{otherwise}\end{array}\right. \tag{22}\]  
where \(r\) is a random value inside the search domain.  
Then, it is possible to define the search domain as it is a closed surface, and the boundary can be written in the following way [36]:  
\[\hat{x}_{i}=\left\{\begin{array}{ll}U_{i}-(L_{i}-x_{i}),&x_{i}<L_{i}\\ L_{i}+(x_{i}-U_{i}),&x_{i}>U_{i}\\ x_{i},&\text{otherwise}\end{array}\right. \tag{23}\]  
This condition is rarely used, but it can improve the optimization if the design variables refer to periodic elements (angles for example).  
The box boundary condition has been analyzed because it can change the convergence proprieties of the algorithm. In particular, in the antenna problem they can highly influence the convergence due to the fact that a physical propriety of the patches, the reflection angle, is characterized by a periodic behavior.  
All the four different functions seen before have been tested on the antenna problem. For each of them, 25 independent trials have been done with 50,000 objective function calls. Figure 5 shows the convergence curves with the four tested feasibility functions.  
Here, it is possible to notice that the eliminating wall has a much worse convergence since the early iterations: this type of condition highly affects the final solution. In fact, the best solution obtained with this condition has no patches with length equal to the minimum or the maximum: in this case, the feasibility condition is reducing the number of solutions  
Figure 5: Comparison among the different convergence curves obtained with the four box boundary conditions. The curves are represented in semilogarithmic scale.  
that can be easily reached by the optimizer. Moreover, this problem requires a good amount of exploitation, especially in the central part of the optimization time.  
The closed search space has an intermediate behaviour, but the convergence is slower than the two best conditions. In fact, the continuous oscillation of the individuals from one side of the domain to the other can create in some parts of the optimization process an alternate attraction on the other individuals, resulting in a slower convergence.  
The other two conditions are characterized by a very similar convergence. Similar considerations can be drawn analyzing the numerical results shown in Table 1.