The properties of restarted genetic algorithms are investigated. The objective is to maximize strictly positive function on a set of integers \(\{0,1,\dots,2^{L-1}\}\). The method consists of evolving population of fixed size using three mechanisms of random selection, crossover and mutation. The mutation rate is of special interest because slow convergence of the mutation rate to 0 can result to jumping away from good solutions prematurely, while rapid convergence to 0 can result, without an appropriate crossover rule, in getting hung up at local extrema.
In the paper, sending the mutation rate to 0 geometrically fast is investigated. A crossover rule called wrong side of the tracks, yields geometric convergence to 0 as \(n \to \infty\) of the probability that the goal state has not been encountered by epoch \(n\). This crossover restarts the algorithm when the progress of improvement in the objective is too slow. It is shown that without the crossover studied here, which amounts to a tempered restart of the algorithm, the asserted geometric convergence need not hold.