Geometric probabilistic evolutionary algorithm

•Probabilistic analysis of geometric transformations leads to new search operators.•The nonlinearity of inversions w.r.t. hyperspheres is imitated stochastically.•Probabilistic search mechanisms inherit geometric transformation properties.•Uniformly distributed reflections add exploration capabilities to an EA.•Scientific contributions show competitive performance in benchmark problems. In this paper we introduce a crossover operator and a mutation operator, called Bernoulli Reflection Search Operator (BRSO) and Cauchy Distributed Inversion Search Operator (CDISO) respectively, in order to define the search mechanism of a new evolutionary algorithm for global continuous optimisation, namely the Geometric Probabilistic Evolutionary Algorithm (GPEA). Both operators have been motivated by geometric transformations, namely inversions with respect to hyperspheres and reflections with respect to a hyperplanes, but are implemented stochastically. The design of the new operators follows statistical analyses of the search mechanisms (Inversion Search Operator (ISO) and Reflection Search Operator (RSO)) of the Spherical Evolutionary Algorithm (SEA). From the statistical analyses, we concluded that the non-linearity of the ISO can be imitated stochastically, avoiding the calculation of several parameters such as the radius of hypersphere and acceptable regions of application. In addition, a new mutation based on a normal distribution is included in CDISO in order to guide the exploration. On the other hand, the BRSO imitates the mutation of individuals using reflections with respect to hyperplanes and complements the CDISO. In order to evaluate the proposed method, we use the benchmark functions of the special session on real-parameter optimisation of the CEC 2013 competition. We compare GPEA against 12 state-of-the-art methods, and present a statistical analysis using the Wilcoxon signed rank and the Friedman tests. According to the numerical experiments, GPEA exhibits a competitive performance against a variety of sophisticated contemporary algorithms, particularly in higher dimensions.