bfgs

Use BFGS method to compute quasi-hessians

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		damped	Numerical safeguarding for BFGS updates

Description

Broyden-Fletcher-Goldfarb-Shanno (BFGS) update will be used to compute quasi-Hessians.

$$B_{k+1}=B_k-\frac{B_k{s}_k{s}_k^T{B}_k}{s_k^T{B}_k{s}_k}+\frac{y_k{y}_k^T}{y_k^T{s}_k}$$

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 $|y_k^T{s}_k|<{10}^{-6}{s}_k^T{B}_k{s}_k$

• Additional safeguarding can be added using the damped option, which utilizes an alternative damped BFGS update when the curvature condition $y_k^T{s}_k>0$ is nearly violated.