PROC NLP: Hessian and CRP Jacobian Scaling :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

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The NLP Procedure

Hessian and CRP Jacobian Scaling

The rows and columns of the Hessian and crossproduct Jacobian matrix can be scaled when using the trust region, Newton-Raphson, double dogleg, and Levenberg-Marquardt optimization techniques. Each element $g_{i,j}$ , is divided by the scaling factor , where the scaling vector is iteratively updated in a way specified by the HESCAL= option, as follows:

No scaling is done (equivalent to

).

$i \neq 0$

First iteration and each restart iteration:

$d_i^{(0)} = \sqrt{\max(| g^{(0)}_{i,i}|,\epsilon)}$

refer to Moré (1978):

$d_i^{(k+1)} = \max ( d_i^{(k)},\sqrt{\max(| g^{(k)}_{i,i}|,\epsilon)} )$

refer to Dennis, Gay, and Welsch (1981):

$d_i^{(k+1)} = \max ( 0.6 d_i^{(k)},\sqrt{\max(| g^{(k)}_{i,i}|,\epsilon)} )$

is reset in each iteration:

$d_i^{(k+1)} = \sqrt{\max(| g^{(k)}_{i,i}|,\epsilon)}$

where $\epsilon$ is the relative machine precision or, equivalently, the largest double precision value that when added to 1 results in 1.

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