Threshold selecting: best possible probability distribution for crossover selection in genetic algorithms

The paper considers the problem of selecting individuals in the current population in Genetic Algorithms for crossover to find a solution of high fitness of a given combinatorial optimization problem. Many different schemes have been considered in literature as possible selection strategies, such as Windowing, Exponential reduction, Linear transformation or normalization and Binary Tournament selection. It is shown that if one wishes to maximize any linear function of the final state probabilities, e.g. the fitness of the best individual of the final population of the algorithm, then the best probability distribution for selecting individuals in each generation is a rectangular distribution over the individuals sorted by their fitness values. This means uniform probabilities have to be assigned to a group of the best individuals of the population but probabilities equal to zero to individuals with fitness ranks higher than a fixed cutoff, which is equal to a certain rank in the sorted fitness vector. The considered strategy is called Threshold Selecting. The proof applies basic arguments of Markov chains and linear optimization and makes only a few assumptions on the underlying principles and hence applies to a large class of Genetic Algorithms.