# Optimization Environment Definition for Beam Steering Reflectarray Antenna Design
## 3 Evolutionary Algorithms
### Algorithm Comparison  
The defined optimization procedure is flexible to be adapted to different EAs. Here, the algorithm previously described has been compared on the optimization of the beam scanning RA. The population size has been set for each algorithm according to a preliminary sensitivity analysis on a smaller antenna problem. The population sizes for each algorithm are reported in Table 2. For what concerns all the other algorithm parameters, they have been set according to a sensitivity analysis performed on Schwefel-226 function because it is the benchmark in which the algorithm has a behavior closer to the antenna problem. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Algorithm** & **DE** & **GA** & **SGA** & **mBBO** & **nBBO** & **PSO** & **SNO** \\ \hline Population size & 25 & 50 & 25 & 25 & 25 & 25 & 100 \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical results of the comparison. Figure 6: Analysis of scalarization factor \(\lambda\) for the four scan angles: for each plot, the left axis shows the average values of REs (solid line) and the confidence interval of 90% (light area). Similarly, the right axis represents the same values for the \(\Delta\theta_{s}\). (**a**) Results for \(\theta_{s}=10^{\circ}\); (**b**) Results for \(\theta_{s}=20^{\circ}\); (**c**) Results for \(\theta_{s}=30^{\circ}\); (**d**) Results for \(\theta_{s}=40^{\circ}\). In order to have a fair comparison between algorithms, the number of objective function calls has been used as termination criterion: in fact, for problems in which the system behavior calculation is computationally expensive, this value is proportional to the total optimization time [37]. In this specific case, the total required time for one trial is around 1h, while the self time of the algorithms ranges from 2 s to 3 s. The selected number of objective function calls is 50,000 because it guarantees an optimal trade-off between the quality of the final results and the optimization time. In this comparison, 25 independent trials have been done for each algorithm in order to guarantee a statistical reliability of the obtained results. These results are reported in Table 3, in which the average value of the independent trials, the standard deviation, and the best results are reported. In Table 3, the best results obtained have been indicated again with the bold font. The most performative algorithms are DE and SNO: the first one is able to achieve the best solution over the different trials, while the second one has the best mean value and the lowest standard deviation. These two values confirm the very good reliability of the optimization with SNO: this is a very important aspect for the problem scalability because in the optimization of larger antennas, it is impossible to perform a large number of independent trials. In order to have more information regarding the behaviour of the algorithms, in Figure 7 the average convergence curves are reported with a semi-logarithmic scale. This plot shows that, even if the final results of DE and SNO are quite similar, they convergence behaviour is different. In fact, on one hand, DE has a very fast initial conver \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Mean** & **Standard Deviation** & **Best Result** \\ \hline DE & 196.57 & 67.49 & **125.13** \\ GA & 6124.97 & 1329.65 & 4224.91 \\ SGA & 873.38 & 1778.9 & 234.97 \\ mBBO & 539.64 & 106.72 & 335.32 \\ nBBO & 402.39 & 106.84 & 269.46 \\ PSO & 26,823.4 & 6314.18 & 13,603.59 \\ SNO & **195.95** & **27.65** & 154.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results of the comparison between different EAs. In bold the best values. Figure 7: Average convergence curves for the analyzed algorithms. gence and then the performance improvement is very low, while SNO has a more constant convergence rate. This means that, if more time is given to the algorithms, SNO is more likely to improve its solutions. A deeper investigation has been performed on the convergence curves of the four most promising algorithms.