Convergence Theorems for a Class of Gene Expression Programming and the Generalization
The two entities called genotype/phenotype in gene expression programming algorithms(GEP) provide many advantages different from that of traditional evolutionary algorithms.However,they make trouble for establishing the Markov model.In order to simplify analyses on probability structure of genetic operators,with genotype searching space this study focuses on the convergence of a class of GEP under gentle conditions.At first,for multimodal fitness functions caused by the genotype and phenotype mapping,we reconstruct the transition matrix of the Markov model corresponding to GEP with elitist recording strategy is reconstructed.And then,we prove its convergence in mean,almost sure convergence and even complete convergence,through addressing and bounding an exact expression for the rate of probability convergence.Compared to old conclusions with rigorous assumptions,our model is more applicable to characteristics of the algorithms.Moreover,the convergence results are stronger and the optimal state set is less.In addition,the exact expression for convergence in probability can be generalized to adaptive evolutionary algorithms.