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Broyden Family of Methods and the BFGS Update: BFM

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# Broyden Family of Methods and the BFGS Update

### Article Outline

Keywords

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References

Keywords Unconstrained optimization - BFGS update - DFP update - Broyden family of methods - Rank-two updates - Quasi-Newton methods

*Quasi-Newton methods*attempt to update a Hessian approximation (or the inverse of it) instead of evaluating the Hessian matrix exactly at each iteration, as in the basic Newton method for

*unconstrained optimization*. Consider the optimization problem:

For this problem the Newton method requires the solution and updating iteratively of the solution point according to:

where

where for

(1) |

**H**(*x*^{(k)}) denotes the Hessian matrix at point*x*^{(k)}(*k*th iteration of Newton's method),**g**(*x*^{(k)}) is the gradient vector at the same point, and finally ∆*x*^{(k)}is the correction to the point*x*^{(k)}. The correction is applied according to: