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 \(y_k = \nabla f_{k+1} - \nabla f_k\) 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.