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Broyden Family of Methods and the BFGS Update

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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:
(1)
where H(x (k)) denotes the Hessian matrix at point x (k) (kth 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:
where for