## 2 Wasserstein Barycenters for the Evolutionary Optimization of Sets
### Wasserstein-Based Crossover and Mutation Operators
#### 2.2.2 Mutation  
We introduce below the boundary mutation, which includes points located at the border of the domain in order to dilate the clouds and counteract the previously discussed shrinking effect.  
**Definition 4**.: _The boundary mutation of a cloud of points \(P_{X_{c}}\) is defined as_  
\[P_{X_{m}}=\operatorname*{arg\,min}_{P_{X_{c}}\#X_{c}=\#X_{c}}\epsilon W_{2}^{2 }(P_{X},P_{X_{c}})+(1-\epsilon)W_{2}^{2}(P_{X},P_{X_{c}\cup X_{bound}}),\]  
_where \(X_{bound}\) is a cloud of points at the domain boundary that can be randomly sampled. In the union \(X_{c}\cup X_{bound}\), the repetitions of identical points are allowed. To clarify, for a polygon, \(X_{bound}\) is the union of points randomly sampled on its sides. For this mutation, the size of the support of \(P_{X_{m}}\) is permitted to be equal to the one of \(P_{X_{c}}\)._  
We also introduce a more general mutation, defined as follows:  
**Definition 5**.: _The full domain mutation of a cloud of points \(P_{X_{c}}\) is defined as_  
\[P_{X_{m}}=\operatorname*{arg\,min}_{P_{X_{c}}\#X=m}\epsilon W_{2}^{2}(P_{X}, P_{X_{c}})+(1-\epsilon)W_{2}^{2}(P_{X},P_{X_{rand}}),\]  
_where \(X_{rand}\) is the union of points uniformly sampled in the domain. Its size is chosen randomly with one of the two following schemes:_  
* _in the space of all possible sizes: full size choice_
* _in_ \(\{n-1,n,n+1\}\) _with_ \(n=\#X_{c}\)_: ternary size choice._  
Figure 3: \(X_{1}\) and \(X_{2}\) are two initial clouds and \(X\) represents their equal weight Wasserstein barycenter.  
The number of the resulting clouds of points and their supports' sizes are determined with the same method as the one used in the crossover._  
The proposed boundary mutation can be compared to the Levy flight jumps used in particle swarm optimization (PSO) [40]. Both boundary mutation and Levy flights enable large perturbations of the current solutions. When the random weight \(\epsilon\) is almost null, the produced clouds differ a lot from the clouds before the mutation, and similarly, the Levy flights can result in substantial movement of a given particle. Both mechanisms help avoiding premature convergence, which is a risk coming from the contracting crossover and the elitist population replacement. However, the boundary mutation and Levy flights differ substantially:  
* The weight \(\epsilon\) is uniformly random, whereas Levy flights jumps occur deterministically in case of prolonged stagnation.
* In the space of clouds, the boundary mutation is focused towards expanding the cloud of points, which is a design choice meant to counteract the effect of crossover, whereas the Levy flights have no preferential direction of perturbation.  
Figure 4 illustrates the boundary mutation of an initial cloud (\(X\)) on the left and its full domain mutation on the right. With boundary mutation, the points only move slightly and some of them are dilated outside of the initial cloud, whereas the full domain mutation disrupts the cloud more noticeably and in a way less oriented towards the boundary.  
If \(\#X_{rand}=\#X_{c}\), then \(m=\#X_{c}\). Otherwise, the mutation produces two sets, one with \(m=\#X_{c}\) and one with \(m=\#X_{rand}\). The two (Boundary and Full Domain) mutations can be arranged in the different ways, as explained below.