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Remarks on Basic Strategy:

We use the grid to ensure the procedure is robust; with additional refinements of the grid we are assured that asymptotically the sequence of designs {xc} converges to a stationary point of f under certain mild assumptions [6,7,10].
Ours is a feasible point method (since it has been our experience that simulations often will fail at infeasible points--and still may fail at feasible points [1,2]); any feasible xc may be chosen to start the search and only feasible candidates xt with be considered.
We can use any of a wide variety of approximation techniques.
We favor those that are amenable to updates; in particular, we use the kriging techniques favored in the design and analysis of computer experiments (DACE) literature [13] because these are the approximation techniques with which we have some experience.
For simplicity, we use cubic splines for the illustrative examples that follow. (We note that spline interpolation is mathematically equivalent to kriging.)
We can use any of a wide variety of optimization methods to produce a candidate xt.
We have used both quasi-Newton methods [11] and pattern search methods [2] to find a candidate xt.
For simplicity, we use the ``eyeball'' method for the examples that follow.
Again we stress that we insist on a candidate xt that is on the grid to ensure convergence to a stationary point of f.
The interpolatory approximations we have favored to date allow us to incorporate f(xt) in a straightforward fashion, though we will have more to say about potential difficulties.
The theory requires strict improvement of f at xc to prevent ``cycling.'' We can confirm xc is a local stationary point of f at the current resolution of the grid by evaluating f at the 2p adjacent grid points defined by the positive and negative coordinate directions.
Again, this point should be made clearer in the examples that follow.
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Virginia Torczon