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(10,000/42) to place a single point in the "90 percent region." The same breeding plan performs as well or better on the 13 other test functions. Given the variety of the test functions, the simplicity of the basic algorithms, and the restricted data base, this is a striking performance. |
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Daniel Frantz concentrated on the internal workings of genetic plans, observing the effect, upon the population, of dependencies in the performance function. Specifically, he studies situations in which the quantity to be optimized is a function of 25 binary parameters. I.e., e consists of functions which are 25-dimensional and have a domain of 225 = 3.2 X 107 points. Dependencies between the parameters (nonlinearities) are introduced to make it impossible to optimize the functions dimension by dimension (unimodality is avoided). Frantz's procedure is to detect the effects of these dependencies upon population structure (gene associations) by using a multidimensional chi-square contingency table. As expected from theoretical considerations (see Lemma 7.2 and the discussion following it) algebraic dependencies (between the parameters) induce statistical dependencies (between alleles). That is, in the population, combinations of alleles associated with dependent parameters occur with probabilities different from the product of the probabilities of the individual alleles. Moreover there is a positional effect on the rate of improvement: For functions with dependencies the rate of improvement is significantly greater when the corresponding alleles are close together in the representation. This effect corresponds to the theoretical result that the ability to pass good combinations on to descendants depends upon the combinations' immunity to disruption by crossover. It is significant that, for the problems studied, the optimum was attained in too short a time for the inversion operator to effectively augment the rate of improvement (by varying positional effects). |
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The results presented in this book have a bearing on several problem areas substantially more difficult than those recounted in section 9.1. Each of these problems has a long history and is complex enough to make sudden resolution unlikely. Nevertheless the general framework does help to focus several disparate results, providing suggestions for further progress. |
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As a first example, let us look at the complex of problems concerned with the dynamics of speciation. These problems have their origin in biology, but a close look shows them to be closely related to problems in the optimal allocation of |
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