BFGS formula

The BFGS formula differs from the DFP equation by an additional term ... 

The one-step BFGS formula is usually more efficient than the Fletcher-Reeves method. It uses somewhat more complex formulas ... 

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... 

The BFGS formula is generally preferred to (26) since computational results have shown that it requires considerably less effort, especially when inexact line searches are used. Quasi-Newton methods, also referred to as variable metric methods, are much more widely used than either the steepest descent or Newton s method. For additional details and computational comparisons, see Fletcher (1987, pp. 44-74). 

Between-operations analysis (methods engineering), 1374-1385 BFGS formula, 2552 Bias(es) ... 

Note that, as B is symmetric, it corresponds to its transpose. The BFGS formula to update B is... 

In this case, the BFGS formula may be simplified to avoid the product BiAx, ... 

Practical tests showed that the BFGS formula often allows a better prediction of the minimum with respect to the other formulae therefore, it usually requires a small amount of iterations and is, thus, the most widely used in practice. 

The updating of the Hessian of the Lagrange function is often computationally ( ) infeasible because of matrix filling due to the BFGS formula. Conversely, the updating of each Hessian with Schubert s formula minimizes the overall matrix filling. 

In many programs based on the SQP method, the Hessian is updated starting from a very simple symmetric matrix (often, I) and using the BFGS formula to guarantee that the approximation of the matrix V L was positive definite. 

Since the matrices It are updated by the BFGS-formula (step 7), the... 

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... 

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