Introduction:

• 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 useful introduction.

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:

1.

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).

2.

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.)

3.

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.