sr1

Use the Symmetric Rank 1 update method to compute quasi-Hessians

Specification

• Alias: None

• Arguments: None

Description

The Symmetric Rank 1 (SR1) update (specified with the keyword sr1) will be used to compute quasi-Hessians.

$$B_{k+1}=B_k+\frac{(y_k-B_k{s}_k)(y_k-B_k{s}_k{)}^T}{(y_k-B_k{s}_k{)}^T{s}_k}$$

where $B_k$ is the $k^{th}$ approximation to the Hessian, $s_k=x_{k+1}-x_k$ is the step and $y_k=\mathrm{\nabla }{f}_{k+1}-\mathrm{\nabla }{f}_k$ is the corresponding yield in the gradients.

Notes

• Initial scaling of $\frac{y_k^T{y}_k}{y_k^T{s}_k}I$ is used for $B_0$ prior to the first update.

• Numerical safeguarding is used to protect against numerically small denominators within the updates.

• This safeguarding skips the update if $|(y_k-B_k{s}_k{)}^T{s}_k|<{10}^{-6}||s_k|{|}_2||y_k-B_k{s}_k|{|}_2$