### Restricting the Step Length

Almost all line-search algorithms use iterative extrapolation techniques that can easily lead them to (feasible) points where the objective function f is no longer defined or is difficult to compute. Therefore, PROC NLMIXED provides options restricting the step length or trust region radius , especially during the first main iterations.

The inner product of the gradient and the search direction is the slope of along the search direction . The default starting value in each line-search algorithm () during the main iteration k is computed in three steps:

1. The first step uses either the difference of the function values during the last two consecutive iterations or the final step-size value of the last iteration to compute a first value of .

• If the DAMPSTEP option is not used,

with

This value of can be too large and can lead to a difficult or impossible function evaluation, especially for highly nonlinear functions such as the EXP function.

• If the DAMPSTEP=r option is used,

The initial value for the new step length can be no larger than r times the final step length of the former iteration. The default value is r = 2.

2. During the first five iterations, the second step enables you to reduce to a smaller starting value by using the INSTEP=r option:

After more than five iterations, is set to .

3. The third step can further reduce the step length by

where u is the maximum length of a step inside the feasible region.