# Quantum Selection for Genetic Algorithms Applied to Electromagnetic Design Problems
## 7 Results and Discussion
### Convergence Speed  
The convergence speed represented by the SRMF is shown in Figure 6 as well as its MA obtained as explained in the Section 7.1. Since the SRMF fluctuates too quickly, the discussion of the results is focused on the MA. For the MTF, three regions are present again. They are described as follows:  
* Early generations (EGs): \[\begin{cases}\text{MA of SRMF}_{\text{GAQS SBX}}\geq\text{MA of SRMF}_{\text{GA}}\\ \text{MA of SRMF}_{\text{GAQS SBX}}\geq\text{MA of SRMF}_{\text{GAQS}}\\ \end{cases};\]
* Middle generations (MGs): MA of SRMF\({}_{\text{GAQS SBX}}\) \[\geq\text{MA of SRMF}_{\text{GA}};\]
* Late generations (LGs): MA of SRMF\({}_{\text{GAQS}}\geq\text{MA of SRMF}_{\text{GA}}\) \[\geq\text{MA of SRMF}_{\text{GAQS SBX}}.\]  
In EG, the observed behavior and the reasoning behind it are the same as the ones presented in Section 7.1. In MG, the MA of SRMF provides more information on the behavior observed before. The way GAQS catches up with GA and GAQS SBX in fitness is clearly explained by GAQS having a slower convergence speed in the beginning but keeping it higher for longer. On the other hand, GA and GAQS SBX start to decelerate only after a few generations. This behavior can be explained by the higher genetic diversity offered by QCO and a decrease in the performance of SBX after the initial exploration.  
In LG, the decrease in performance is even more noticeable when GA starts to outperform GAQS SBX in convergence speed. The decrease in performance of SBX is caused by excessive search outside the boundaries set by the parents and an increase in the performance of HCO near the minimum in the MTF. Once again, the moment when the regions appear is completely problem-dependent.  
For the EMDP only two regions are clearly distinguishable. However, their characteristics vary with respect to the MTF and they are defined as  
* Initial generations (IGs): \[\begin{cases}\text{MA of SRMF}_{\text{GAQS SBX}}\geq\text{MA of SRMF}_{\text{GA}}\\ \text{MA of SRMF}_{\text{GAQS SBX}}\geq\text{MA of SRMF}_{\text{GAQS}}\\ \end{cases};\]
* Final generations (FGs): \[\begin{cases}\text{MA of SRMF}_{\text{GAQS}}\geq\text{MA of SRMF}_{\text{GA}}\\ \text{MA of SRMF}_{\text{GAQS}}\geq\text{MA of SRMF}_{\text{GAQS SBX}}.\end{cases}\]  
For both EMDPs, the IG is marked by a clear difference in the initial convergence speed between GAQS SBX and the other algorithms. This is a showcase of the positive impact of SBX in early generations, where its search capacities are fully exploited.  
For the microwave filter in FG, the search capacity of SBX becomes counterproductive after some generations when the heuristic crossover starts to explore the region delimited by the parents more efficiently. Nonetheless, the convergence speed of GAQS SBX is still higher than that of GA due to the resistance to stagnation offered by QSO.  
On the other hand, in FG for the EM absorber, GAQS SBX has on average the slowest convergence speed, but the overhead in fitness obtained in IG is too significant, explaining its decline with respect to GA. Nonetheless, GAQS keeps a higher convergence speed than GA, showcasing the advantage of keeping a higher genetic diversity to both convergence speed and resistance to stagnation.