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Our illustrations are necessarily simple in nature--and the results shown are better than we would expect for many (most?) problems in higher dimensions. Nonetheless, we would hope that two things are now clear:
that there is value in using the optimization to guide when and where to sample the objective to build an algebraic approximation and
that this should be done so in a way that pays some heed to the overall quality of the approximation.
Again, we stress that this latter observation is not unique to us. Recent work from the nonlinear programming community [3,9] has emphasized the need to monitor the "geometry" of the sample points even when using them to build a local quadratic approximation of the objective--in particular, to avoid the numerical ill-conditioning that can arise if such considerations are ignored. And ideas current in the global optimization community [4,5,8] have stressed the need to adopt strategies that sample in regions where relatively little is known about the objective in an effort to devise heuristics that are more likely to produce global minimizers.

In higher dimensions, we are not sanguine that good approximations of the objective can be constructed over the entire design region. The curse of dimensionality is likely to hold sway for all but the simplest of objective functions. Nonetheless, we wish to avoid the tendency of most derivative-based optimization techniques to find the local minimizer nearest the initial trial point used to start the optimization process. Furthermore, by attempting to construct a global rather than a local approximation to the objective, we can use such tools as analysis of variance (ANOVA) to discern other information about the objective either before or after the optimization process.

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Virginia Torczon