### Restricting the Step Length

Almost all line-search algorithms use iterative extrapolation techniques which can easily lead them to (feasible) points where the objective function is no longer defined. (e.g., resulting in indefinite matrices for ML estimation) or difficult to compute (e.g., resulting in floating point overflows). Therefore, PROC NLP 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 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 length value of the last iteration to compute a first value of .

• Not using the DAMPSTEP= option:

with

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

• Using the DAMPSTEP= option:

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

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

After more than five iterations, is set to .

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

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