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 $G_{i,j}$ , $i,j=1,\ldots ,n,$ is divided by the scaling factor $d_ i \times d_ j$ , where the scaling vector $d=(d_1,\ldots ,d_ n)$ is iteratively updated in a way specified by the HESCAL= $i$ option, as follows:

No scaling is done (equivalent to $d_ i=1$ ).
First iteration and each restart iteration:

$d_ i^{(0)} = \sqrt {\max (|G^{(0)}_{i,i}|,\epsilon )}$
refer to Moré (1978):

$d_ i^{(k+1)} = \max \left( d_ i^{(k)},\sqrt {\max (|G^{(k)}_{i,i}|,\epsilon )} \right)$
refer to Dennis, Gay, and Welsch (1981):

$d_ i^{(k+1)} = \max \left( 0.6 d_ i^{(k)},\sqrt {\max (|G^{(k)}_{i,i}|,\epsilon )} \right)$
$d_ i$ 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.