Hessian and CRP Jacobian Scaling

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The NLP Procedure

Hessian and CRP Jacobian Scaling

The rows and columns of the Hessian and cross-product 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, ... ,n, is divided by the scaling factor d_i * d_j, where the scaling vector d = (d₁, ... ,d_n) is iteratively updated in a way specified by the HESCAL=i option, as follows:

i = 0

: No scaling is done (equivalent to d_i=1).

$i \neq 0$

: First iteration and each restart iteration sets:

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

i = 1

: see Mor $\acute{e}$ (1978):

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

i = 2

: see Dennis, Gay, & Welsch (1981):

$d_i^{(k+1)} = \max(.6 * d_i^{(k)},\sqrt{\max(| G^{(k)}_{i,i}|,\epsilon)})$

i = 3

: 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.

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