Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation

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Abstract
This paper aims at establishing fundamental theoretical properties for a class of ''genetic algorithms'' in continuous space (GACS). The algorithms employ operators such as selection, crossover, and mutation in the framework of a multidimensional Euclidean space. The paper is divided into two parts. The first part concentrates on the basic properties associated with the selection and mutation operators. Recursive formulae for the GACS in the general infinite population case are derived and their validity is rigorously proven. A convergence analysis is presented for the classical case of a quadratic cost function. It is shown how the increment of the population mean is driven by its own diversity and follows a modified Newton's search. Sufficient conditions for monotonic increase of the population mean fitness are derived for a more general class of fitness functions satisfying a Lipschitz condition. The diversification role of the crossover operator is analyzed in Part II [1]. The treatment adds much light to the understanding of the underlying mechanism of evolution-like algorithms.

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