Population Fixed-Points for Functions of Unitation

The dynamical systems model for the simple genetic algorithm due to N'ose [12] can be simplified to the ca,se of zero crossover, and to fitness functions that divide the search space into relatively few equivalence classes. This produces a low-dimensional system for which the fixed-point can be calculated; it is the leading eigenvector of the system. This technique, applied elsewhere [11] to Royal Road functions, is adapted in this paper to apply to ftmctions of imitation. Infinite population fixed-points are calculated for some simple examples, including trap functions that have previously been analysed in terms of deception [2]. The effects of eigenvectors outside the population space are explored, and finite population behaviour is examined in this way. Two surprising theorems are demonstrated for infinite populations; that every population distribution is the fi.xed-point for some fitness function; and that fitness functions exist for which evolution goes "backwards" towards the global minimum. Theoretical results are backed up with experiments using real, finite population genetic algorithms.