# Population Symmetrization in Genetic Algorithms
## 2 Method Description
### Genetic Algorithm with Symmetrization Operator (GASO)
Two variants of the genetic algorithm with the symmetrization operation were considered, depending on the crossover operator used (similarly to the reference algorithm):  
* GASO--with the default crossover operator (cube crossover);
* GASOSC--with segment crossover. As an example the function No. 10 from the BBOB benchmark [42] was used. This is an ellipsoidal function described by the formula:  
\[f_{10}(\textbf{x})=\sum_{i=1}^{D}10^{\delta\frac{i-1}{D-1}}z_{i}^{2}+f_{\text {opt}} \tag{14}\]  
Its characteristic feature is elongated ellipsoidal contour lines. The 3D plot and the level plot are shown in Figure 2.  
The population points over the first few generations, for the reference algorithm with segment crossover (GASC) and for the corresponding algorithm with the symmetrization operator (GASOSC), for the function No. 10 are presented in Figure 3.  
It can be noticed in Figure 3b that symmetrization concentrates the population along the lowest contours of the objective function. Moreover, as it is clearly visible for generations 4 and 6, the population in the symmetrization algorithm is more evenly distributed around the leader. This enables a more effective search of the \"vicinity\" of the best contour lines using the segment crossover operator. Thus, the symmetrization operator in the proposed GASOSC algorithm has the features of a local search algorithm, but its main feature is the cooperation with the main (genetic) algorithm for the effective operation of the crossover operator. Figure 2: The ellipsoidal function, function no. 10 from the benchmark. 2D plots: surface (**a**) and level (**b**). Black arrow indicates the optimal point of the function. Let us try to justify a bit more formally the meaning of the symmetrization operator in the discussed algorithm. Let us assume that the function \(f\) is convex on some convex surroundings of the point where the function \(f\) reaches a local (global) minimum. Let \(A\),  
Figure 3: The scattering of points (black dots) in the search space in several initial generations (in consecutive rows) for the reference algorithm with segment crossover—GASC (column (**a**)) and for the corresponding algorithm with the symmetrization operator—GASOSC (column (**b**)) on the example of function No. 10 from the set of test functions from BBOB benchmark in dimension 2. Color lines show the contours of the function. be parental points that belong to that environment. For a child point \(C=\alpha A+(1-\alpha)B\), where \(\alpha\in\ \langle 0;\ 1\rangle\), the inequality holds:  
\[f(C)\leq\alpha f(A)+(1-\alpha)f(B) \tag{15}\]  
It follows that \(f\ (C)\ \leq f\ (A)\) or \(f\ (C)\ \leq f\ (B)\), i.e., the child point obtained as a result of segment crossover, is better than at least one of the parents. In the final phase of the program course (exploitation), the best points in the population are generally the points closest to the leader. Applying the symmetrization operator to these points (at the expense of the \"worst\" points) concentrates the population around the leader. This gives a more precise minimization of the objective function as a result of using the crossover operator after symmetrization. During the initial generations, symmetrization of points remote from the leader located in different attraction pools than the leader supports a global search, along with the crossing operation.