Quasi-Newton variable metric

Quasi-Newton, variable metric, conjugate gradient, Fletcher-Powell, Davidon-Fletcher-Powell, Murtagh-Sargent, Broyden-Fletcher-... 

Quasi-Newton or Variable Metric or Secant Methods... 

The quasi-Newton or variable-metric methods introduced by Davidon 1 have now become the standard methods for finding an unconstrained minimum of a differentiable function f(x), and an excellent review of the basic theory has been given by Dennis and Mord f2. ... 

Another technique for handling the mass balances was introduced by Castillo and Grossmann (1). Rather than convert the objective function into an unconstrained form, they implemented the Variable Metric Projection method of Sargent and Murtagh (32) to minimize Gibbs s free energy. This is a quasi-Newton method which uses a rank-one update to the approximation of H l, with the search direction "projected" onto the intersection of hyperplanes defined by linear mass balances. 

The Quasi-Newton, or variable metric, methods of optimization... 

To obtain equilibrium geometries for small molecules and clusters we have implemented a variable metric method which is based on a quasi-Newton scheme and is widely used in optimization theory (Lipkowitz and Boyd 1993 Schlegel 1987). In this... 

Calculation of the inverse Hessian matrix can be a potentially time-consuming operation that represents a significant drawback to the pure second derivative methods such as Newton-Raphson. Moreover, one may not be able to calculate analytical second derivatives, which are preferable. The quasi-Newton methods (also known as variable metric methods) gradually build up the inverse Hessian matrix in successive iterations. That is, a sequence of... 

A brief description of optimizations methods will be given (also see refs. 41-44). In contrast to other fields, in computational chemistry great effort is given to reduce the number of function evaluations since that part of the calculation is so much more time consuming. Since first derivatives are now available for almost all ab initio methods, the discussion will focus on methods where first derivatives are available. The most efficient methods, called variable metric or quasi-Newton methods, require an approximate matrix of second derivatives that can be updated with new information during the course of the optimization. Some of the more common methods have different equations for updating the second derivative matrix (also called the Hessian matrix). 

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). 

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