The presentation that follows is intentionally tutorial in nature.
It is meant to illustrate, by means of some simple examples, two things:
Those familiar with these issues will undoubtedly prefer to move
directly to the paper---a summary of which can be found in the HTML
- Our fundamental perspective that when one uses algebraic
approximations within an iterative optimization process, during the
initial stages of the optimization one should concentrate on the
predictive ability of the surrogate and only worry about
accuracy in the vicinity of a minimizer or when it becomes
clear that the approximation is not doing a good job of identifying
trends in the objective.
- Our experience also suggests there is value in a trade-off
between the single-minded pursuit of a minimizer and the construction
of an approximation that gives a reasonable "picture" of the behavior
of the objective--particularly when a problem is known to have
many local minimizers.
Those less familiar
with these concerns may find the presentation that follows to be a
The Technical Issues:
We begin by noting that neither of the two observations made above is
unique to us. What we hope to do, however,
is use this presentation to motivate the three technical proposals
contained in our paper.
Specifically, our paper addresses the following three issues we have
encountered when using approximations to accelerate an iterative
engineering design optimization process:
- We advocate the introduction of so-called merit
functions that explicitly recognize the desirability of improving the
current approximation to the expensive simulation in certain regions
(i.e., a measure to balance the expenditure of expensive evaluations
of the objective between the search for a single minimizer and the
desire to learn more about the design space).
- We include suggestions for addressing the ill-conditioning that
plagues kriging approximations constructed from sequential designs
generated by optimization algorithms. (This is an issue that is difficult to
illustrate with a simple example, but which we have seen arise in practice.)
- We propose methods for obtaining space-filling designs for
nonrectangular regions and propose pattern search algorithms for such
regions--developments that are new to both the kriging approximation
and the nonlinear optimization literatures. Given that our ultimate
goal is to tackle problems in which the constraints are implicitly
defined by expensive simulations, the need for such extensions is evident.
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