Broyden-Fletcher-Goldfarb-Shanno BFGS formula

The smallest update consistent with (5.37) is achieved 1 the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula, which ensures that if is positive-definite (e.g., = I), so... 

The methods differ in the formula used to generate the sequence S, k=0,l,2,., and after Fletcher and Powell s 3 analysis of Davidon s method a whole spate of formulae were invented in the sixties. Broyden 4 introduced some rationalization by identifying a one-parameter family, and recommended a particular member, now commonly referred to as the BFGS (Broyden-Fletcher-Goldfarb-Shanno) formula. Huang 5 widened the family, but by the end of the sixties numerical experience was producing a consensus that the BFGS formula was the most robust of the formulae available. The formula is... 

One of the most successful and widely used updating formulas is known as BFGS for its four developers Broyden, Fletcher, Goldfarb, and Shanno.6 95 It is a rank 2 update with inherent positive-definiteness (i.e., Bk positive-definite Bk+j positive-definite) that was derived bj symmetrizing the Broyden rank 1 update.5 6 95 A sequence of matrices B is generated from a positive-definite B0 (which may be taken as the identity) by the BFGS formula... 

There are several other methods for updating Dj in step 5. Another important method is based on the BFGS formula, which was developed by Broyden (1970), Fletcher (1970), Goldfarb (1970), and Shanno (1970) and is given by... 

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