Consider the sequence of proposed solutions using the interpolation coupled with the optimization. We denote by xc the current best solution for the "true" problem f(x) and by xt the "trial" solution proposed by the approximation a. Alternatively, xt may be a "confirmation point" if the approximation suggests that a minimizer has already been identified.
At Iteration 8 we stop with a confirmed (local) minimizer of f (at least to the resolution of the grid) at x* = 0.150 and a decent approximation a to our objective f on the interval [ 0, 0.5 ].
It is often said that if a model of the objective or a
constraint function has a weakness, the optimization procedure will
find it. Here we see the same phenomenon using
approximations of the objective. In the early stages of the
optimization process, the approximation predicts a minimizer in
precisely those region(s) where we have the least information about
the objective: consider Iterations 3, 4, and 5. But by using an
iterative process, we can improve the approximation with the eventual
goal of using it to successfully predict a minimizer of f.
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