BFGS-update

One of the most efficient and widely used updating formula is the BFGS update. Broyden (1970), Fletcher (1970), Goldfarb (1970), and Shanno (1970) independently published this algorithm in the same year, hence the combined name BFGS. Here the approximate Hessian is given by... 

Both difficulties are eliminated by replacing V L by a positive-definite quasi-Newton (QN) approximation B, which is updated using only values of L and V L (See Section 6.4 for a discussion of QN updates.) Most SQP algorithms use Powell s modification (see Nash and Sofer, 1996) of the BFGS update. Hence the QP subproblem becomes... 

Some of the most important variations are the so-called Quasi-Newton Methods, which update the Hessian progressively and therefore economize compute requirements considerably. The most successful scheme for that purpose is the so-called BFGS update. For a detailed overview of the mathematical concepts, see [78, 79] an excellent account of optimization methods in chemistry can be found in [80]. 

It is often desirable that the approximate Hessian is positive definite so that the quadratic model has a minimum. To ensure this we may use the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update given by... 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

It is for this reason the positive definite BFGS update Eq. (3.35) is preferred over the PSB update Eq. (3.33) for minimizations. The positive definite Newton step is usually a better direction than steepest descent since it takes into account features of the function further away from xc than does steepest descent. 

In test calculations [75] this algorithm was found to produce rapid convergence. When combined with single-step (in = 1) BFGS update of the inverse Hessian, this is a very efficient algorithm. 

The molecular model of deoxycytidine is examined in Tables 2 and 3 to analyze the issues of preconditioning and number of LM-BFGS updates. Energy model details are described elsewhere,22 23 and here it suffices to note that several well-separated local minima correspond to different feasible combinations of the sugar s pseudorotation parameter and the glycosyl (sugar-to-base orientation) dihedral angle. 

Solution methods for optimization problems that involve only continuous variables can be divided into two broad classes derivative-free methods (e.g., pattern search and stochastic search methods) and derivative-based methods (e.g., barrier function techniques and sequential quadratic programming). Because the optimization problems of concern in RTO are typically of reasonably large scale, must be solved on-line in relatively small amounts of time and derivative-free methods, and generally have much higher computational requirements than derivative-based methods, the solvers contained in most RTO systems use derivative-based techniques. Note that in these solvers the first derivatives are evaluated analytically and the second derivatives are approximated by various updating techniques (e.g., BFGS update). 

Of course, what is really required is Hk+1 = F k+i- But, assuming Hk = Fk has already been obtained, one can use a standard (but messy) formula for inverses of rank two perturbations to invert the expression for Fic+i- Happily in this case most of the mess cancels out, and one arrives at the BFGS update (where again the subscripts k on the right side have been omitted) ... 

This BFGS update is only slightly more complicated than the DFP update, and this very slight extra computation is more than compensated for by the extra speed of convergence that is observed in practice. 

Although one is the complementary formulation of the other, it is worth noting that DFP and BFGS are not equivalent. In fact, the BFGS update for B produces, by taking the inverse, a corresponding update for H, in the form... 

Practical tests confirmed that the BFGS updating procedure is the best of the rank-2 alternatives when one-dimensional searches are limited. Also for this reason, this formula is currently the most recommended of all the available ahematives. 

The inverse of the Hessian matrix is approximated by a symmetric positive definite matrix that is constructed iteratively. To that end, the Scilab function optim () [10] uses the Broyden, Fletcher, Goldfarb and Shanno (BFGS) update [11], Alternatively, the Levenberg-Marquardt algorithm implemented in the Scilab function IsqrsolveO may be used. 

At present, the Broyden-Fletcher-GoIdfarb-Shanno (BFGS-) update... 

There is a close relationship between the DFP- and the BFGS-update. If... 

Gradient converged in 3 cycles Hessian Updated using BFGS Update... 

One of the most successful QN formulas in practice is associated with the BFGS method (for its developers Broyden, Fletcher, Goldfard, and Shanno). The BFGS update matrix has rank 2 and inherent positive-definiteness (i.e., if B is positive definite then Bj + i is positive definite) as long as yjsk < 0. This condition is satisfied automatically for convex functions but may not hold in general. In practice, the line search must check for the descent property updates that do not satisfy this condition may be skipped. 

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