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of different orders can be set adjacent to each other in a string without changing their significance. In consequence, under the combined effect of inversion and reproduction, bits defining various regions of above-average values for f(x) will be ever more tightly linked. This in turn increases the rate of exploration of intersections and refinements of these regions. Filling in the remaining details to complete the extension of the illustration is a straightforward exercise. Section 7.3 in the next chapter provides a detailed example of the response of an algorithm of type  to nonlinearities. Theorem 7.4 of that chapter, coupled with the comments on dimensionality in chapter 4 (p. 71) shows that, whatever the form of f (i.e., for any f mapping a bounded d dimensional space into the reals), an algorithm of type optimizes expeditiously. Moreover, the algorithm does this while rapidly increasing the average value of the points it tests (though they may be scattered through many different hyperplanes), thus making the algorithm useful for "online" control. Sections 9.1 and 9.3 provide more detailed summaries of these advantages. |
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