## 3 Numerical Analysis
### Experimental Protocol
#### 3.2.1 Algorithms Settings  
For all the experiments, the algorithms' hyperparameters are set as follows:  
* We consider square search domains. The length of each side is 100.
* A discretization of step 1 is applied to each side for the sampling on the boundaries for boundary mutation. Continuous alternatives, however, should also work.
* \(n_{min}=10\), \(n_{max}=20\), \(d=2\).
* The population size is set equal to 10 times the average number of active components in each set, \(\mu=10d(1/(n_{max}-n_{min}+1))\sum_{i=n_{min}}^{n_{max}}i=300\), and the maximum number of iterations of the algorithms is \(N_{iter}=500\).
* Regarding the choice of \(\sigma^{2}\), we consider it proportional to \(\mathbb{E}\|X-X^{\prime}\|^{2}\), where \(X\) and \(X^{\prime}\) are points sampled uniformly in the domain. The latter quantity is the mean squared distance between two points sampled uniformly in the domain. In a centered-squared domain of side \(L\), we obtain \(\mathbb{E}\|X-X^{\prime}\|^{2}=4*Var(U)\) with \(Var(U)\), the variance of a uniform continuous law. In our case, \(\sigma^{2}=0.01\mathbb{E}\|X-X^{\prime}\|^{2}=0.01*4*L^{2}/12=33.33\).
* The default value of prob (Algorithm 2), if not specified, is 0.5.  
We denote the proposed evolutionary algorithm, with Wasserstein barycenter-based operators, as Wasserstein-barycenter-based generator evolutionary algorithm (WBGEA). The different versions of this algorithm that we are studying are identified with specific names and presented in Table 1.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Algorithm** & **Crossover** & **Mutation** \\ \hline WBGEA\_1 & Equal Weight crossover (Equation (2)) & Alternating mutation (Algorithm 2). Full size choice (Definition 5). \\ \hline WBGEA\_2 & Equal Weight crossover (Equation (2)) & Successive independently weighted mutation (Algorithm 3). Full size choice (Definition 5). \\ \hline WBGEA\_3 & Equal Weight crossover (Equation (2)) & Successive equally weighted mutation (Algorithm 4). Full size choice (Definition 5). \\ \hline WBGEA\_1t (t for ternary) & Equal Weight crossover (Equation (2)) & Alternating mutation (Algorithm 2). Ternary size choice (Definition 5). \\ \hline WBGEA\_1t\_nc (nc for no crossover) & No crossover & Alternating mutation (Algorithm 2). Ternary size choice (Definition 5). \\ \hline WBGEA\_1t\_rc (rc for random weight in crossover) & Random Weight crossover (Equation (3)) & The mutation is carried out with Algorithm 2. Ternary size choice (Definition 5). \\ \hline Ref\_alg & Uniform crossover on sequences (Algorithm 5) & Gaussian mutation (Algorithm 6) \\ \hline Ref\_alg\_nc & No crossover & Gaussian mutation (Algorithm 6) \\ \hline Ref\_wass & Uniform crossover on sequences (Algorithm 5) & Alternating mutation (Algorithm 2). Ternary size choice (Definition 5). \\ \hline Wass\_gauss & Equal Weight crossover (Equation (2)) & Gaussian mutation (Algorithm 6) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Acronyms of the algorithms studied and names of their evolution operators.  
Comparing specific versions of these algorithms with each other sheds some light on specific questions. These experiments and the questions addressed are detailed in Table 2.  
The test functions used in order to perform the previously described experiments are presented in Section 3.3.