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We will assume that f and satisfy the following assumptions:
 1.
 f is twice continuously differentiable
 2.

 3.
 is symmetric positive definite
The Newton methods update the current iteration by the formula
 
(1) 
where is given by a line search that ensures sufficient decrease.
QuasiNewton methods update an approximation of the Hessian as the iterations
progress. A possible update is the BFGS method Broyden (1969); Fletcher (1970); Goldfarb (1970); Shanno (1970), which
overcomes some limitations of the earlier Broyden's method Broyden (1965).
In particular, the Broyden's update does not keep the spd structure of the Hessian.
This structure not only ensures the existence of a local minimizer but also allows the convergence
of the updated solution to the minimum Kelley (1999).
The BFGS update is a ranktwo update given by
 
(2) 
with and .In practice, it is very useful to express the previous equation in terms of the inverse matrices. We then have
 
(3) 
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Stanford Exploration Project
4/27/2000