Replicators, Majorization and Genetic Algorithms: New Models and Analytical tools

This paper establishes a strong connection between evolutionary algorithms and majorization theory, using replicator models as a bridge. The relationship between replicator selection systems and majorization theory suggests new selection operators, convergence results and theoretical gains such as the availability of convergence results from the well developed theory of inhomogeneous doubly stochastic Markov chains. We also give sufficient conditions under which multiplicative models of crossover implement the majorization pre-ordering, complementing the classical results due to Moran and Nishimura. Quantities such as relative entropy are known to increase during crossover; we provide bounds on the increase. Our techniques apply to a wide class of functions (Schur-convex), and generalize previous results on quadratic dynamical systems. We also present some new results on replicator systems; these include a new interpretation of the necessary and sufficient conditions under which a replicator systems is a gradient descent system, and an application of a variant of Kolmogorov’s representation theorem to yield a generalized fundamental theorem.