Unconstrained Minimization Techniques - Convergence Criteria

Once a new q is found which has reduced the discrepancy function by a reasonable amount, the whole cycle is repeated until at least one of several convergence criteria are met:

1. The discrepancy function is extremely close to zero.

2. An iteration fails to reduce the discrepancy function by more than a very small percentage. In SEPATH, a "relative function change" criterion is checked on each iteration. The criterion on the k'th iteration is

(87)

3. The "residual cosine" criterion of Browne (1982, eq. 1.9.5) falls below a specified tolerance.

One generally finds with path models that the more elements there are in q, the more difficult it is to find the actual q which minimizes the discrepancy function. Thus, all other things being equal, you might expect to need more iterations to converge to a solution for a large problem than for a small one. Unfortunately, the larger the problem, the longer each iteration tends to take. Consequently, good initial estimates can be very important for large problems. Indeed, without good initial estimates, even the best "state of the art" non-linear optimization routine will fail to find solutions for some problems.