# BFGS method

### From Glossary

This is a method to solve an unconstrained nonlinear program that proceeds as follows.

- Start with any symmetric, negative definite matrix, say , and any initial point . Set
- Direction: Solve .
- Step size: .
- Change in position: .
- New point: .
- Change in gradient: .
- Update to form with the BFGS update.
- Let to complete the iteration.

Compare this with the DFP method, and note a key difference is that DFP estimates the inverse hessian, whereas BFGS estimates the hessian (and solves the linear system). There is a correspondance, and empirical evidence suggests, that the BFGS method has less of a problem with error.