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Simple Tests

We show some simple one-dimensional examples for illustrative purposes.

In the sequence of graphs that display our results, the objective f is shown in blue, while the approximation a is shown in cyan.

The grid upon which we are operating is shown in ``hatch'' marks along the x-axis

The points at which f(x) is known are denoted with red circles.

When we have only two known points (which is what we use in the initial experimental design) we build a simple linear interpolant. We distinguish these two initial design sites with yellow stars.

When we have three known points (after the first optimization step), we build a quadratic interpolant.

For subsequent iterations we use a cubic spline interpolant [12], which for the one-dimensional examples we show requires a minimum of four data points.

The candidate xt for which the approximation a predicts decrease in f--and at which we will evaluate f before proceeding--is denoted with a magenta star. When the approximation does not predict any further decrease in f, the magenta star indicates the point at which we will evaluate f (if unknown) to confirm that xc is a local stationary point of f on the grid.

Warning: The pictures that follow were generated using Matlab. The sequence of graphs presented have been scaled by Matlab to best portray the range of values specific to that graph; thus, different scales for the y-axis are used for different graphs to capture the changing features of the interpolant as additional points are added. (The scale of the x-axis remains constant.)

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Virginia Torczon