Updating formula

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... 

If the BFGS algorithm is applied to a positive-definite quadratic function of n variables and the line search is exact, it will minimize the function in at most n iterations (Dennis and Schnabel, 1996, Chapter 9). This is also true for some other updating formulas. For nonquadratic functions, a good BFGS code usually requires more iterations than a comparable Newton implementation and may not be as accurate. Each BFGS iteration is generally faster, however, because second derivatives are not required and the system of linear equations (6.15) need not be solved. 

The Newton-Raphson scheme prescribes the updating formula... 

Dealing with Z BZ directly has several advantages if n — m is small. Here the matrix is dense and the sufficient conditions for local optimality require that Z BZ be positive definite. Hence, the quasi-Newton update formula can be applied directly to this matrix. Several variations of this basic algorithm... 

Line searches are often used in connection with Hessian update formulas and provide a relatively stable and efficient method for minimizations. However, line searches are not always successful. For example, if the Hessian is indefinite there is no natural way to choose the descent direction. We may then have to revert to steepest descent although this step makes no use of the information provided by the Hessian. It may also be... 

For example, a matrix of rank 1 can be formulated as the outer product of two vectors, such as uvT. (The rank of this matrix is 1 because all rows are scalar multiples of one other). Applying condition [46] to this update form B +1 = Bk + uvT, we obtain the condition that u is a vector in the direction of (yk - Bksk). If yk = Bksk,Bk already satisfies the QN condition [46]. Otherwise, we can write the general rank 1 update formula as ... 

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... 

Equation (26) is the "Quasi-Newton" condition it is fundamental to all the updating formula. There have been many updates proposed, and we briefly review some of the more important ones. The simplest are based on... 

There is a whole class of rank two update formula, the two most important members are... 

An alternative type of update formula has been suggested by Greenstadt [33]. The updating formula has the usual form... 

The augmented Hessian method requires an exact Hessian, or an update method on the Hessian itself. The update formula for the Hessian analysis to the inverse Hessian appear in the Appendix. 

TABLE VI NUMBER. OF OPTIMIZATION CYCLES UPDATE FORMULAE. REQUIRED FOR DIFFERENT... 

Here we consider the relative efficiencies of the different inverse update formula. The calculations take the initial inverse Hessian as a unit matrix and the initial in the line... 

The various SAHN methods differ in the way in which the proximity between clusters is defined in step 1 and how the merged pair is represented as a single cluster in step 3. The proximity calculation in step 3 typically uses the Lance-Williams matrix-update formula ... 

Table 1 Parameter Values for Some Common SAHN Methods Defined by the Lance-Williams Matrix Update Formula"...

The computational effort in evaluating the Hessian matrix is significant, and quasi-Newton approximations have been used to reduce this effort. The Wilson-Han-Powell method is an enhancement to successive quadratic programming where the Hessian matrix, (q. ), is replaced by a quasi-Newton update formula such as the BEGS algorithm. Consequently, only first partial derivative information is required, and this is obtained from finite difference approximations of the Lagrangian function. 

Note that, since L has units (m/sf, the nonnegative function h ) would be dimensionless. With this model for A the realizability condition in Fq. (B.52) would always yield a nonzero upper bound on At when h ) is finite. physically, E is null in the limit of pure particle trajectory crossing where the true NDF is a sum of Dirac delta functions. On the other hand, when E reaches its maximum value, the NDF is Gaussian. Thus, since mixed advection is associated with random particle motion, the model in Fq. (B.56) also makes physical sense. Nonetheless, the potential for singular behavior in the update formula makes the treatment of mixed advection problematic. 

Eq. (18) is symmetric and positive definite (i.e. the eigenvalues of the Hessian are all positive), and minimizes the norm of the change in the Hessian. Corresponding updating formulae also exist for the inverse of the Hessian [68-71,77], which allow the algorithm to avoid the inversion of H needed in Eq. (17). 

If Broyden s updating formula is used, it is possible to avoid the solution of the... 

An alternative to factorization upgrading is Broyden s updating formula applied to the inverse matrix ... 

This relation represents a hyperplane in the space of unknowns dfj/dx Projecting the point with coordinates (dfj/dxk on this hyperplane, the correction with the minimum Euclidean norm is obtained and, in turn, the updating formula for the components to be mmierically evaluated is... 

Schubert s updating formula is particularly useful in the case of large-scale ( ) sparse systems, since it preserves Jacobian sparsity and executes only the calculations for nonzero derivatives. 

The Subject Index is a two-yearly cumulation covering both Volume 43 and 44, and is supported by an updated Formula Index of Complex Functional Groups (p. 493) for rapid access to functional group nomenclature (the formula OSiCj, for example, indicating that the function RjSiO —C = C is indexed as Enoxysilanes in the Subject Index). 

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