Abstract

. Genetic algorithms are now among the most promising optimization techniques. They are based on the following reasonable idea. Suppose that we want to maximize an objective function J(x). We somehow choose the first generation of "individuals" x 1 ; x 2 ; :::; x n (i.e., possible values of x) and compute the "fitness" J(x i ) of all these individuals. To each individual x i , we assign a survival probability p i that is proportional to its fitness. In order to get the next generation we then repeat the following procedure k times: take two individuals at random (i.e., x i with probability p i ) and "combine" them according to some rule. For each individual of this new generation, we also compute its fitness (and survival probability), "combine" them to get the third generation, etc. Under certain reasonable conditions, the value of the objective function increases from generation to generation and converges to a maximal value. The performance of genetic algorithms can be essentially improved if we use fitness scaling, i.e., use f(J(x i )) instead of J(x i ) as a fitness value, where f(x) is some fixed function that is called a scaling function. The efficiency of fitness scaling essentially depends on the choice of f . So what f should we choose? In the present paper we formulate the problem of choosing f as a mathematical optimization problem and solve it under different optimality criteria. As a result, we get a list of functions f that are optimal under these criteria. This list includes both the functions that were empirically proved to be the best for some problems, and some new functions that may be worth trying. 1 1.

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