Threshold-selecting strategy for best possible ground state detection with genetic algorithms

Genetic algorithms are a standard heuristic to find states of low energy in complex state spaces as given by physical systems such as spin glasses but also in combinatorial optimization. The paper considers the problem of selecting individuals in the current population in genetic algorithms for crossover. Many schemes have been considered in literature as possible crossover selection strategies. We show for a large class of quality measures that the best possible probability distribution for selecting individuals in each generation of the algorithm execution is a rectangular distribution over the individuals sorted by their energy values. This means uniform probabilities have to be assigned to a group of the individuals with lowest energy in the population but probabilities equal to zero to individuals which are corresponding to energy values higher than a fixed cutoff, which is equal to a certain rank in the vector sorted by the energy of the states in the current population. The considered strategy is dubbed 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 algorithms.