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The NLMIXED Procedure

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 difficult to compute. Therefore, PROC NLMIXED provides options restricting the step length \alpha or trust region radius \Delta, especially during the first main iterations.

The inner product gTs of the gradient g and the search direction s is the slope of f(\alpha) = f(\theta + \alpha s)along the search direction s. The default starting value \alpha^{(0)} = \alpha^{(k,0)} in each line-search algorithm ( \min_{\alpha \gt 0} f(\theta + \alpha s) )during the main iteration k is computed in three steps:

  1. The first step uses either the difference df=|f(k) - f(k-1)| of the function values during the last two consecutive iterations or the final step-size value \alpha^{\_} of the last iteration k-1 to compute a first value of \alpha_1^{(0)}.
  2. During the first five iterations, the second step enables you to reduce \alpha_1^{(0)} to a smaller starting value \alpha_2^{(0)} using the INSTEP=r option:
    \alpha_2^{(0)} = \min (\alpha_1^{(0)},r)
    After more than five iterations, \alpha_2^{(0)} is set to \alpha_1^{(0)}.
  3. The third step can further reduce the step length by
    \alpha_3^{(0)} = \min (\alpha_2^{(0)},\min(10,u))
    where u is the maximum length of a step inside the feasible region.

The INSTEP=r option enables you to specify a smaller or larger radius \Delta of the trust region used in the first iteration of the trust region and double dogleg algorithms. The default initial trust region radius \Delta^{(0)} is the length of the scaled gradient (Mor\acute{e} 1978). This step corresponds to the default radius factor of r=1. In most practical applications of the TRUREG and DBLDOG algorithms, this choice is successful. However, for bad initial values and highly nonlinear objective functions (such as the EXP function), the default start radius can result in arithmetic overflows. If this happens, you can try decreasing values of INSTEP=r, 0 < r < 1, until the iteration starts successfully. A small factor r also affects the trust region radius \Delta^{(k+1)} of the next steps because the radius is changed in each iteration by a factor 0 \lt c \le 4, depending on the ratio \rho expressing the goodness of quadratic function approximation. Reducing the radius \Delta corresponds to increasing the ridge parameter \lambda, producing smaller steps directed more closely toward the (negative) gradient direction.

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