Again we consider the sequence of proposed solutions using our merit
function coupled with the optimization. As before, we denote by
*x*_{c} the current best solution for
the "true" problem *f*(*x*). But now
*x _{t}* is the trial solution proposed by the merit
function

- Iteration 1:
*x*_{c}= 0.20,*x*= -0.50_{t} - Iteration 2:
*x*_{c}= 0.20,*x*= 0.50_{t} - Iteration 3:
*x*_{c}= 0.20,*x*= 1.50_{t} - Iteration 4:
*x*_{c}= 0.20,*x*= -0.20_{t} - Iteration 5:
*x*_{c}= 0.20,*x*= 1.20_{t} - Iteration 6:
*x*_{c}= 1.20,*x*= 1.30_{t} - Iteration 7:
*x*_{c}= 1.30,*x*= 1.40_{t} - Iteration 7:
*x**= 1.30

It is the case that because we now assign "merit" to
sampling in regions for which we have no information about the
objective, we are slower to improve the best known solution to the
optimization problem. For this example, *x*_{c} is not
replaced until Iteration 6 whereas, earlier, working only with the
approximation *a*, we replaced *x*_{c} by
Iteration 3.

On the other hand, using the merit function---rather than just
the approximation--enforces a better "spread" of the sample points
throughout the design region. The
beneficial effects of this are apparent by Iteration 5. Earlier, when
we did not make use of the merit function, with the exception of the
two end points, all our sampling occurred in the
basin with a local minimizer at 0; thus, the
behavior of the objective in the rest of the design space is largely
unknown. However, when we use the merit function to select candidate
designs, by Iteration 5 even the *approximation* indicates that
a global minimizer should be in the region near 1.2; the merit function
*m* only further emphasizes this prediction.

Note that the approximation *a* still predicts the local
minimizer at 0--and the merit function *m* emphasizes this
prediction too. If we wished to search for other minimizers, we
certainly have adequate predictive ability to do so. But now the
global minimizer dominates. Having identified it, we are content to
stop.

** Next:** Summary
** Previous:** A Family of Merit Functions