BFGS-method

The alkaline solution containing 2,4-D and 2,4,5-T residues was acidified and extracted with diethyl ether. NH4OH (0.1 ml) was added to the extract which was then taken to dryness. The residue was butylated with 14% BCI3 in n-BuOH following the concentrated-BFg method described by Woolson and Harris (2). 

The BFGS method is considered to be superior to DFP in most cases because (a) it is less prone to loss of positive definiteness or to singularity problems... 

Solution. Using the Optimization Toolbox from MATLAB, the BFGS method requires 20 iterations before the search is terminated, as shown below. 

Liu, D. C., and Nocedal, J., On the limited memory BFGS method for large scale optimization, Tech. Rep. NAM 03, Northwestern University (1988). 

Diese BFg-Methode der Kondensation unsymmetrischer Ketone mit Saureanhydriden zu a-monosubstituierten -Diketonen stellt eine wertvolle Erweiterung der experimentellen Methodik dar, da sich die -Diketone ihrer geringen CH-Aciditat wegen im allgemeinen nach den ublichen Methoden nur schwer a-alkylieren lassen. 

D. C. Liu and J. Nocedal, Math. Prog., 45, 503 (1989). On the Limited Memory BFGS Method for Large Scale Optimization. 

We notice that even at optimization, none of these procedures yield particularly accurate Hessian matrices. For most of the systems we have examined, however, the BFGS method is most accurate, followed by the rank one method. Least accurate is the Greenstadt method. The modes least accurately represented are the softer modes. If this system had been given an initial geometry that had C3v symmetry, then only the 2a modes 3 and 4 of the tables, would be updated. 

The computational work involves mostly dot products. If the maximum number of updates is fixed, the work scales linearly. The L-BFGS method has been used in a number of optimization problems in computational chemistry [62,64,89,90]. 

A local induced fit motion is sampled by ADAM docking in combination with an energy minimization. ADAM utilizes a vdW-offset 3D grid. To enlarge the pocket uniformly, the potential curve is shifted that allows the atoms to get closer. The stmcture optimization covers side chains and ligand movements. The energy minimization is performed with the L-BFGS method [74]. 

As Bi+i is to be symmetric, and z, are selected to assure symmetry. In the BFGS method, they are assigned equal to Ag and BjAx , leading to... 

Note that rtk = 0 yields the DFP method and = 1 the BFGS method. The optimally conditioned (OC) method chooses to minimize the condition number of the Hessian (the condition number is the ratio of the largest to the smallest eigenvalue), thereby improving the behavior of the optimization. The CG, MS and DFP methods are also special cases of the Huang family of algorithms. Equations (7)-(10) can also be used to update the Hessian, B, rather than its inverse, H, provided that Ax and Ag are interchanged when H is replaced by B. ... 

It is important to do local optimizations as efficiently as possible because they are a time-consuming part of minima hopping. In our implementation, we perform local minimization in two steps. The first optimization uses the limited memory L-BFGS method [64, 65] with a loose convergence threshold. The optimization is refined in the second step which uses Davidon s optimally conditioned variable metric method [66] and a more stringent convergence criterion. 

Br(>yden-Fletcher-G[Pg.192]

Remark 1 Step 5 must still be specified. The matrix has been introduced to indicate that the matrix k is modified by a (low-rank) correction matrix. When step 5 is reset by a specific update formula, the quasi-Newton method is named after that update (for instance BFGS-method, DFP-method, Broyden-method,. ..). 

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