- 1.
- (a)
- 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 {*x*_{c}}
converges to a stationary point of *f* under certain mild
assumptions
[6,7,10].
- (b)
- 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
*
x*_{c} may be chosen to start the search and only
feasible candidates *x*_{t} with be considered.

- 2.
- 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.)
- 3.
- (a)
- We can use any of a wide variety of optimization methods to
produce a candidate
*x*_{t}.

We have used both quasi-Newton methods [11] and pattern
search methods [2] to find a candidate
*x*_{t}.

For simplicity, we use the ``eyeball'' method for the
examples that follow.

Again we stress that we insist on a candidate *x*_{t}
that is on the grid to ensure convergence to a stationary point of
*f*.
- (b)
- The interpolatory approximations we have favored to date allow
us to incorporate
*f*(*x*_{t}) in a
straightforward fashion, though we will have more to say about
potential difficulties.
- (c)
- The theory requires strict improvement of
*f* at *
x*_{c} to prevent ``cycling.'' We can confirm
*x*_{c} is a local stationary point of *f* at
the current resolution of the grid by evaluating *f* at the
2*p* adjacent grid points defined by the positive and
negative coordinate directions.

Again, this point should be made clearer in the examples that follow.

** Next:** Simple Tests
** Previous:** Basic strategy:

*Virginia Torczon*

*6/25/1998*