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:
- 1.
- that there is value in using the optimization to guide when and
where to sample the objective to build an algebraic approximation and
- 2.
- 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.
Next: Outstanding Issues:
Previous: Discussion of a Family of Merit Functions:
Virginia Torczon
6/23/1998