## 4 Results and Discussions
### Relaxed Crossover
#### 4.3.2 Role of the Boundary Mutation  
The following results further investigate the role of the boundary mutation within the alternating Wasserstein mutation (Algorithm 2). This mutation carries out a boundary mutation with a probability of prob or a full domain mutation with a probability of 1-prob. The different values of prob studied are 1, 0.5, 0.1, 0.05 and 0 within the WBGEA_1t_nc algorithm (i.e., there is no crossover). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Algorithm** & \(\mathbf{F_{0}}\) & \(\mathbf{F_{90}}\) & \(\mathbf{F_{45}}\) & \(\mathbf{F_{4d}}\) & \(\mathbf{F_{minDist}}\) & \(\mathbf{F_{inert}}\) \\ \hline WBGEA\_1 & 89.158 (\(\pm\)0.196) & 89.183 (\(\pm\)0.133) & 89.714 (\(\pm\)0.193) & 87.908 (\(\pm\)0.157) & 31.785 (\(\pm\)0.530) & 71,112 (\(\pm\)700) \\ \hline WBGEA\_2 & 88.147 (\(\pm\)0.148) & 88.279 (\(\pm\)0.139) & 88.704 (\(\pm\)0.112) & 86.936 (\(\pm\)0.137) & 31.279 (\(\pm\)0.455) & 67,451 (\(\pm\)462) \\ \hline WBGEA\_3 & 88.117 (\(\pm\)0.219) & 88.033 (\(\pm\)0.140) & 88.476 (\(\pm\)0.132) & 86.672 (\(\pm\)0.129) & 30.977 (\(\pm\)0.467) & 63,510 (\(\pm\)538) \\ \hline WBGEA\_1t & 89.270 (\(\pm\)0.163) & 89.334 (\(\pm\)0.154) & 89.872 (\(\pm\)0.208) & 87.908 (\(\pm\)0.157) & 31.892 (\(\pm\)0.404) & 76,480 (\(\pm\)566) \\ \hline Ref\_alg & 87.281 (\(\pm\)0.438) & 87.345 (\(\pm\)0.408) & 87.398 (\(\pm\)0.319) & 85.742 (\(\pm\)0.307) & 31.923 (\(\pm\)1.337) & 81,869 (\(\pm\)758) \\ \hline Ref\_alg\_nc & 88.734 (\(\pm\)0.208) & 88.681 (\(\pm\)0.281) & 88.891 (\(\pm\)0.222) & 87.453 (\(\pm\)0.179) & **35.780** (\(\pm\)0.495) & 79,562 (\(\pm\)737) \\ \hline Ref\_was & 88.938 (\(\pm\)0.293) & 89.036 (\(\pm\)0.378) & 88.906 (\(\pm\)0.265) & 86.974 (\(\pm\)0.228) & 30.769 (\(\pm\)0.572) & **94,211** (\(\pm\)616) \\ \hline Wass\_gauss & 88.430 (\(\pm\)0.242) & 88.383 (\(\pm\)0.257) & 88.667 (\(\pm\)0.191) & 87.199 (\(\pm\)0.166) & 35.108 (\(\pm\)0.857) & 79,116 (\(\pm\)1575) \\ \hline WBGEA\_1t\_nc & **89.939** (\(\pm\)0.237) & **89.928** (\(\pm\)0.183) & 89.895 (\(\pm\)0.161) & **88.269** (\(\pm\)0.185) & 34.561 (\(\pm\)0.525) & 82,375 (\(\pm\)484) \\ \hline WBGEA\_1t\_nc (prob = 1) & 89.879 (\(\pm\)0.251) & 89.823 (\(\pm\)0.211) & **89.952** (\(\pm\)0.231) & 87.997 (\(\pm\)0.220) & 34.635 (\(\pm\)0.578) & 85,947 (\(\pm\)625) \\ \hline WBGEA\_1t\_nc (prob = 0.1) & 89.612 (\(\pm\)0.212) & 89.698 (\(\pm\)0.189) & 89.737 (\(\pm\)0.105) & 88.065 (\(\pm\)0.160) & 33.051 (\(\pm\)0.742) & 74,948 (\(\pm\)521) \\ \hline WBGEA\_1t\_nc (prob = 0.05) & 89.346 (\(\pm\)0.205) & 89.399 (\(\pm\)0.216) & 89.438 (\(\pm\)0.111) & 87.846 (\(\pm\)0.113) & 32.447 (\(\pm\)641) & 71,907 (\(\pm\)898) \\ \hline WBGEA\_1t\_nc (prob = 0) & 88.195 (\(\pm\)0.201) & 88.166 (\(\pm\)0.182) & 88.316 (\(\pm\)0.124 & 86.921 (\(\pm\)0.216) & 30.986 (\(\pm\)0.491) & 51,306 (\(\pm\)1195) \\ \hline WBGEA\_1t\_rc & 89.649 (\(\pm\)0.191) & 89.613 (\(\pm\)0.220) & 89.686 (\(\pm\)0.168) & 88.097 (\(\pm\)0.166) & 32.785 (\(\pm\)0.655) & 78,888 (\(\pm\)604) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean (with 5 significant digits) \(\pm\) std. deviation of the best values returned by the algorithms, calculated over 20 independent repetitions for each test function. For each function, the best results are boldfaced. We observe on the left of Figures 21-23 that when we diminish the probability of performing the boundary mutation (prob) from 0.5 to 0, the performances of the algorithms (speed and optimality) decrease on all the test functions. However, a maximal value of prob is not the optimal configuration of WBGEA_1t_nc since when prob = 1, the performance of \(F_{4d}\) is inferior to the one with prob = 0.5. The diversity varies in the opposite way of the performances (right plot of Figures 21-23), except for WBGEA_1t_nc with prob = 0 and prob = 1. The small chances of boundary mutation favor the contracting property stated in Theorem 1 of the Wasserstein barycenter in the full domain mutation.