Analysis of the Simple Genetic Algorithm on the Single-peak and Double-peak Landscapes

We compare the behavior of a GA with and without crossover. A simple GA with crossover can have two stable fixed points (bistability) on the single-peak landscapes for string lengths at least 8. There are catastrophic transitions from one to two and back to one stable fixed point as the mutation rate increases. Crossover also causes the fixed-point population distribution to change. For a fixed point near the peak, there are fewer copies of the optimum string, more copies of near-optimum strings, and less copies of far-from-optimum strings. An example is given where average population fitness decreases to the minimum possible population fitness for the with-crossover GA, while the average population fitness increases for the without-crossover GA. The primary tool in obtaining these results is the Vose dynamical system model.