## 5 Conclusions and Perspectives  
We have proposed new evolutionary algorithms made to optimize functions defined over sets of points. The mutation and crossover operators are based on Wasserstein barycenters. The performances of the algorithms were tested on a family of test functions including wind-farm layout emulators. We have proven that the Wasserstein barycenter contracts the points within the convex hull of existing points. To counteract this effect, we have introduced the boundary mutation that biases the Wasserstein barycenters towards the domain boundaries. Our experiments have shown that because Wasserstein-based operators better preserve the geometrical structure of point sets than traditional sequence-based operators, they allow performance gains: replacing a classical operator with its Wasserstein equivalent always improved the search. As a side result, we have found that on our point set test suite, the crossovers are, on average, detrimental to the search. The designs returned by the Wasserstein-based evolutionary algorithm were satisfactory in that they made physical sense: the best wind-farm layouts spread the turbines over the entire design domain and had alignments perpendicular to dominating wind directions; the designs for inertia had points grouped at the corners of the domain; the designs for maximin point distance had uniformly spread points. Of course, these solutions, which were generated by stochastic (evolutionary) optimizers, would benefit from a final fine-tuning with a local search.  
This work would benefit from three short term continuations. First, the domain of definition of the points in the set might be non convex. For example, one could consider domains with a hole inside, as happens with overland wind farms. The current operators, through the Wasserstein barycenters, will create points in such a hole and need to be upgraded.  
Another continuation is to investigate the effectiveness of the algorithm for clouds of points of dimensions greater than two.  
Third, when dealing with computationally expensive functions over point sets, the proposed evolutionary algorithm could be included within a Bayesian optimization algorithm, where it would allow us to optimize the acquisition criteria.  
A longer term continuation to this work is to extend it to clouds of colored points. An example would be to optimally select the technology of the wind turbines (the color) within a catalog while optimizing the farm layout.  
Conceptualization, B.S., R.L.R., J.P., M.K. and S.Z.; methodology, B.S., R.L.R. and J.P.; software, B.S.; formal analysis, B.S., R.L.R. and J.P.; writing--original draft preparation, B.S.; writing--review and editing, B.S., R.L.R., J.P., M.K. and S.Z.; visualization, B.S.; supervision, R.L.R., J.P., M.K. and S.Z.; project administration, R.L.R., M.K., J.P. and S.Z. All authors have read and agreed to the published version of the manuscript.  
This work was funded in part by the French Agence Nationale de la Recherche through the SAMOURAI project, ANR-20-CE46-0013.  
Not Applicable.  
The pseudo-code of the algorithms and the test functions are described in the paper. The hyper-parameters of the algorithms are also given. For any question for future use, please contact authors.  
Authors [PERSON], [PERSON] and [PERSON] were employed by the company EDF R&D. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.