Quasi-Newton methods examples

EXAMPLE 5.1 COMPARISON OF NEWTON, FINITE DIFFERENCE NEWTON, AND QUASI-NEWTON METHODS APPLIED TO A QUADRATIC FUNCTION... 

Newton and quasi-Newton methods are used for more difficult convergence problems, for example, when there are many recycle streams, or many recycles that include operations that must be converged at each iteration, such as distillation columns. The Newton and quasi-Newton methods are also often used when there are many recycles and control blocks (see Section 4.8.1). The Newton method should not normally be used unless the other methods have failed, as it is more computationally intensive and can be slower to converge for simple problems. 

Cmbined methds. There are numerous other methods in the literature for finding transition states. However, the more common methods use simpler numerical algorithms in a more efficient way. The Berny optimization algorithm and the synchronous transit quasi-newton method (STQN) are good examples. 

By utilizing forces on FES, we can identify the SS and TS stmctures in solution with full optimization with respect to all coordinates of the solute molecules. For example, if we adopt the quasi-Newton method with the following Broyden-Fletcher-Goldfarb-Shanno (BFGS) procedure [26-29] for stractural optimization scheme in the FEG method, the i -F l)-th reactant structure is taken as,... 

In Chapter 4 the Gauss-Newton method for systems described by algebraic equations is developed. The method is illustrated by examples with actual data from the literature. Other methods (indirect, such as Newton, Quasi-Newton, etc., and direct, such as the Luus-Jaakola optimization procedure) are presented in Chapter 5. 

True quasi-Newton schemes attempt to sidestep any calculation of the exact Hessian. In such cases, an approximate Hessian must be estimated by the program or by the user. How should such an estimate be obtained One attractive possibility is to utilize experimental data on vibrational frequencies, or to transfer estimates from one system to a related system. Here is where the choice of coordinate system can be crucial. In general, such estimates are likely to provide information only about, in effect, the normal modes of vibration of a system they specify only the diagonal elements of the Hessian in a particular coordinate system. There are few sources of empirical information about coupling force constants, for example, and few sources about force constants expressed in Cartesian coordinates. Also Hessian information in Cartesian coordinates is rarely transferable from one molecule to another. Hence, for first-order methods the choice of coordinate system in which the optimization is performed is strongly influenced by the need to obtain Hessian information. 

Local optimisation methods include, for example, the steepest descent, Newton, quasi-Newton, conjugate gradient or simplex methods. Important concepts in local optimisation wiU be presented below. 

For the optimization of the energy (10.7.69), we may in principle apply any scheme developed for the unconstrained minimization of multivariate functions - for example, some globally convergent modification of the Newton method or some quasi-Newton scheme. Expanding the energy to second order by analogy with (10.1.21), we obtain... 

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