Optimization

We restrict ourselves to minimization and essentially unconstrained problems, i.e., problems where the minimizers are in the interior of a box H. We recognize the two separate problems local optimization and global optimization.

Local optimization means finding the local minimum detectable from a given starting point. Global optimization means finding the lowest minimum in the whole of H.

Local optimization may be difficult but in global optimization finding the global minimum cannot be guaranteed (finding a solution requires that all points in H are tried unless some knowledge making it possible to reduce this to a finite number is available).

It has been shown (Brent) that the problem of finding the global minimizer is not even a properly posed problem, i.e., there exist continuous functions in H with maximum absolute difference in function values arbitrary small but with global minimizers wide apart.