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

- Iteration 1:
*x*_{c}= 0.167,*x*= 0.0_{t} - Iteration 2:
*x*_{c}= 0.167,*x*= 0.217_{t} - Iteration 3:
*x*_{c}= 0.167,*x*= 0.50_{t} - Iteration 4:
*x*_{c}= 0.167,*x*= 0.067_{t} - Iteration 5:
*x*_{c}= 0.167,*x*= 0.433_{t} - Iteration 6:
*x*_{c}= 0.167,*x*= 0.150_{t} - Iteration 7:
*x*_{c}= 0.150,*x*= 0.133_{t} - Iteration 8:
*x** = 0.150

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*.

** Next:** Second Example
** Previous:** First Example