Every numerical analyst can tell you many examples where the computed result is nowhere near the solution. Computers are used for numerical computation because such situations are rare. But in such a rare case, the result looks just as trustworthy as they usually do: there is no warning. (Experts can tease out warnings, but not always.)
That is one problem. The other problem is that conventional numerical analysis can only find local minima in optimization problems. In problems with an unknown and possibly large number of local minima it is not known how the local minima that are found relate to the global minimum.
The first step towards a solution to the problems of conventional numerical computation is interval arithmetic . This approach is especially interesting for global optimization .
The equality in an equation is only one of several useful constraints. The numerical variables in an equation are only one type of useful variable: variables can be integer, boolean, or range over other domains. Thus, systems of equations are generalized to constraint systems .
In the special case where the variables are numeric, the sets of their possible values are given by intervals. In that case the constraints include equality, less-than-or-equal, and various arithmetic constraints. Then we have interval constraints .
Constraint processing offers new opportunities for massive parallelism.
Q: Will DFO find me a global minimum?
A: Maybe, but the algorithm is designed only to find local minima. In any case the
user should run it from a variety of starting points. Because of the
general approach the method has a tendency not to converge to "poor" local optima.
Exactly. To get a lower bound for the global minimum, when derivatives are not available, one needs interval methods.
Modified December 26, 2002