Convergence Characteristics of Keep - Best

This paper presents some theoretical convergence characteristics of Keep-Best Reproduction (KBR), a selection strategy for genetic algorithms (GAs). We have previously introduced KBR and reported encouraging results in the traveling salesman domain (Wie98a) where KBR was compared with the standard replacement strategy of replacing the two parents by their two children (STDS). Here we demonstrate that in a non-operator environment as well as in the ONEMAX domain KBR has the same convergence characteristics as 2-tournament selection and elitist recombination (ELR) (Thi94a). We also show how a modiication of ELR suggested in (Thi97) can be utilized to tune the selective pressure of KBR. These analytical models are fairly simplistic and cannot accurately model the convergence characteristics in more complex domains where building blocks are correlated, such as the TSP domain. We will give some empirical results of a comparison of KBR and ELR in this domain. We have previously developed a family competition scheme which we call Keep-Best Reproduction (KBR). KBR takes two parents, recombines them and then keeps the best parent and the best oospring in order to introduce good new genetic material into the population as well as to keep good old chromosomes. Intuitively this increases the selection pressure which should lead to faster convergence but by keeping the best parent we hope to maintainenough diversity to avoid overs-election and premature convergence. Other researchers have used the idea of a competition (global or local) as well but their implementations all diier from KBR. A discussion of these other algorithms is not possible within the length of this short paper, but references are given at the end. The selection pressure of a selection scheme is usually quantiied by its selection intensity I: I(t) = S(t) (t) = f s (t) ? f(t) (t) Here the selection diierential S(t) is the diierence between the average tness of the parent population at generation t, f s (t), and the population mean tness at generation t, f(t). Assuming a standardized normal distribution of the initial population's tnesses, i.e. N(f;) = N(0; 1), the selection intensity I simply becomes the expected average tness of the population after applying the selection scheme to the original population. This is exactly the model that Blickle and Thiele used to compute selection intensities (Bli95). They have derived the selection intensity for tournament selection of size s to be 1 p 2 e ? y 2 2 dy s?1 …