The Dynamics of a Genetic Algorithm under Stabilizing Selection

A bstract. A formalism recently int roduced [4, 5] uses the met hods of statist ical mech anics to mod el the dynamics of gene t ic algor it hms (GAs) . To be of mor e general int erest this formalism must be able to descr ibe problems ot her th an the tes t cases cons idered in [5]. In this pap er , the technique is applied to th e subset sum prob lem , which is a combinatorial optimizat ion problem with a strongly nonlinear energy (fit ness) fun ct ion and many local minima under single spin flip dyn ami cs. It is a pr ob lem that exhibits interest ing dyn ami cs, reminiscent of stabilizing select ion in popu lation biology. The dyn amics are solved under certain simplifying ass umptions and are reduced to a set of difference equa tions for a small nu mb er of relevant qu an ti ties. The quantities used are the cumulants of the population , which describe it s sha pe, and th e mean corre lation wit hin the populati on , which measures the microscopic simila rity of populat ion members. Including the mean corre lat ion allows a better description of the popula t ion than the cumulants alone would pr ovide and represents a new and important extension of the technique. The formalism includes finit e population effects and descr ibes problems of realisti c size. T he theory is shown to agree closely to simulations of a real GA and the mean best energy is acc urate ly predict ed .