Discussion of Second Example

We consider this second example to confirm that the first example was not an isolated anomaly and also to set the stage for two of the outstanding issues we propose to address.

Optimization

Again we consider the sequence of proposed solutions using the interpolation coupled with the optimization. We denote by x_c the current best solution for the "true" problem f(x) and by x_t the trial solution proposed by the approximation a.

Iteration 1: x_c = 0.20, x_t = -0.5
Iteration 2: x_c = 0.20, x_t = 0.10
Iteration 3: x_c = 0.10, x_t = 1.50
Iteration 4: x_c = 0.10, x_t = -0.20
Iteration 5: x_c = 0.10, x_t = 0.0
Iteration 6: x_c = 0.0, x_t = -0.10
Iteration 7: x* = 0.0

A particular feature of this example is that by the second iteration we have already identified the minimizer of the current approximation a. We do not assume that this means that we have found a minimizer of f. Instead, to confirm that x_c is a minimizer of f we must look at the two grid points adjacent to x_c. When we evaluate the first of these two confirmation points (in Iteration 2), and incorporate the result in the approximation used at Iteration 3, we see that there is still a great deal of uncertainty about the location of a minimizer of f. However, by the time we reach Iteration 5, the approximation is good enough to predict the true minimizer of f. Again, we must confirm this. We have already evaluated f at one adjacent grid point, 0.10, so we evaluate f at the other adjacent grid point, -0.10.

At Iteration 7 we stop with a confirmed (local) minimizer of f (at least to the resolution of the grid) at x* = 0.0 and a decent approximation a to our objective f on the interval [ -0.5, 1.5 ].

Next: Second Example Revisited Previous: Second Example

Virginia Torczon
6/13/1998