Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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bfgs

Use BFGS method to compute quasi-hessians

Specification

Alias: none

Argument(s): none

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.