# Population Symmetrization in Genetic Algorithms
## 1 Introduction  
Global optimization methods can be divided into two main groups [1]: single-solution metaheuristics and population metaheuristics. The first group includes, among others, the simulated annealing [2], tabu search [3], variable neighborhood search [4], guided local search [5], and iterated local search [6]. The second group emmes evolutionary algorithms inspired by the Darwinian theory of evolution and swarm intelligence, which is based on patterns of the social behavior of animal colonies--mainly insects, birds, and bacteria. Evolutionary algorithms are a fairly broad concept encompassing genetic algorithms [7], evolution strategy [8], and genetic programming [9]. This class also includes differential evolution [10] and cultural algorithms [11]. It turns out that the processes governing cultural change are very similar to the processes of biological evolution. The class of swarm intelligence algorithms covers, among others, ant colony optimization [12], particle swarm optimization [13], bee colony optimization [14] or artificial immune systems [15]. In recent years rapid development of machine learning methods can also be observed in terms of metaheuristics design [16]. The authors focus on genetic algorithms (GA), which are nondeterministic population-based metaheuristics widely used in practice. There are three main areas of the effective application of genetic algorithms: operation management, multimedia, and wireless networks [17]. In operation management the scheduling [18], or inventory control [19] are good examples. Multimedia applications are mainly encryption [20], image processing, and medical imaging. In wireless networks, GA is effective at optimizing routing, load balancing, or channel allocation. Following the no free lunch theorem, there is no one universal optimizer [21]. Despite many advantages of GAs that justify their broad application, they also have disadvantages. These include a slow convergence speed, premature convergence, or difficulty in obtaining high precision of the approximation of global extremes. One way to improve the properties of GAs is to modify the genetic operators. Many variants of the crossover operator for real-coded GA have been proposed over the past decades, for example, heuristics crossover [22], arithmetical crossover [23],direction based crossover [24], direction-based exponential crossover [25], unimodal normal distribution crossover [26] or Laplace crossover [27]. Some research on new mutations [28; 29] and selection operators [30; 31] have been carried out. The second widely used approach, most often in the context of specific applications, is a GA hybridization. In the work [17] a classification of combining GA with other algorithms was made:  
* Enhancing search capability [32; 33];
* Generating feasible solutions [34];
* Replacing of genetic operators [35];
* Optimizing control parameters [36]. In our work, real-coded genetic algorithms used in the problem of optimization of continuous functions are of interest. Limiting the global extremum search area to the \(D\) dimensional hypercube determines the appropriate selection of genetic operators--they cannot lead GA beyond the established search area. It seems, however, that in the classical genetic algorithm, after a certain number of iterations, the generation of points is distributed in a certain multidimensional convex cone whose vertex is the population leader. Child points obtained by crossing points with such a population still remain in this cone. If the searched point realizing the global minimum of the objective function (more precisely--the target optimization area guaranteeing the assumed error) is outside the cone, it is inaccessible for population points obtained by crossing. Mutations give the chance of leaving the cone in the desired direction, but it is a multi-generational process--the algorithm usually prematurely converges and stops earlier.