Newton method quasi

The main difference between the modified Newton and quasi-Newton methods is in the evaluation of the Hessian the modified Newton methods approximate the matrix using local information in the neighborhood of xj the quasi-Newton methods update the matrix using gradient values evaluated at each iteration. 

The quasi-Newton methods have the following advantages. 

1) Heavy analytical or numerical evaluations of the Hessian are prevented. 

2) As the Hessian is updated and not reevaluated at each iteration, it is possible to solve the linear system (3.57) using a more high-performance technique, as discussed later. 

As explained at the beginning of this section, a quasi-Newton method is obtained when in the classical Newton process the Hessian matrix is approximated by estimate matrices which are computed by an update formula. Subsequently a few results pertaining to the behaviour of convergence of some quasi-Newton methods are presented. Furthermore, the applicability of these methods for locating minimizers and/or saddle points of energy functionals is discussed. 

A quasi-Newton method is essentially described by the following steps  

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,. ..). 

Remark 2 Procedure I can also be started with an estimate M to 

In that case the matrix must be reset by (step 2 and 

There are a number of variations on the Newton-Raphson method, many of which aim to eliminate the need to calculate the full matrix of second derivatives. In addition, a family of methods called the quasi-Newton methods require only first derivatives and gradually construct the inverse Hessian matrix as the calculation proceeds. One simple way in which it may be possible to speed up the Newton-Raphson method is to use the same Hessian matrix for several successive steps of the Newton-Raphson algorithm with only the gradients being recalculated at each iteration. 

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 

At each iteration k, the new positions X/t+i are obtained from the current positions x, the gradient and the current approximation to the inverse Hessian matrix H  

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point Xjt and at the point x t+i and determine the minimum in the fitted function. 

The symbol when interposed between two vectors means that a matrix is to be formed. The yth element of the matrix u v is obtained by multiplying u, by Vj. 

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In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation and its analogue = Aq must hold (where - g " and... 

Quantum mechanical calculations are restricted to systems with relatively small numbers of atoms, and so storing the Hessian matrix is not a problem. As the energy calculation is often the most time-consuming part of the calculation, it is desirable that the minimisation method chosen takes as few steps as possible to reach the minimum. For many levels of quantum mechanics theory analytical first derivatives are available. However, analytical second derivatives are only available for a few levels of theory and can be expensive to compute. The quasi-Newton methods are thus particularly popular for quantum mechanical calculations. 

A transition structure is, of course, a maximum on the reaction pathway. One well-defined reaction path is the least energy or intrinsic reaction path (IRC). Quasi-Newton methods oscillate around the IRC path from one iteration to the next. Several researchers have proposed methods for obtaining the IRC path from the quasi-Newton optimization based on this observation. 

Peng, C. and Schlegel, H.B., Combining Synchronous Transit and Quasi-Newton Methods to Find Transition States , Israel Journal of Chemistry, Vol. 33, 449-454 (1993)... 

In HyperChem, two different methods for the location of transition structures are available. Both arethecombinationsofseparate algorithms for the maximum energy search and quasi-Newton methods. The first method is the eigenvector-following method, and the second is the synchronous transit method. 

The synchronous transit method is combined with quasi-Newton methods to find transition states. Quasi-Newton methods are very robust and efficient in finding energy minima. Based solely on local information, there is no unique way of moving uphill from either reactants or products to reach a specific reaction state, since all directions away from a minimum go uphill. 

The Synchronous Transit-Guided Quasi-Newton Method(s)... 

These methods utilize only values of the objective function, S(k), and values of the first derivatives of the objective function. Thus, they avoid calculation of the elements of the (pxp) Hessian matrix. The quasi-Newton methods rely on formulas that approximate the Hessian and its inverse. Two algorithms have been developed ... 

Gill, P.E. and W. Murray, "Quasi-Newton Methods for Unconstrained Optimization", J. Inst. Maths Applies, 9,91-108 (1972). 

Newton and Quasi-Newton Methods of Unidimensional Search.157... 

NEWTON AND QUASI-NEWTON METHODS OF UNIDIMENSIONAL SEARCH... 

In the quasi-Newton method (secant method) the approximate model analogous to Equation (5.7) to be solved is... 

Quasi-Newton methods start out by using two points xP and jfl spanning the interval of jc, points at which the first derivatives of fix) are of opposite sign. The zero of Equation (5.9) is predicted by Equation (5.10), and the derivative of the function is then evaluated at the new point. The two points retained for the next step are jc and either xP or xP. This choice is made so that the pair of derivatives / ( ), and either/ (jc ) or/ ( ), have opposite signs to maintain the bracket on jc. This variation is called regula falsi or the method of false position. In Figure 5.3, for the (k + l)st search, x and xP would be selected as the end points of the secant line. 

Quasi-Newton methods may seem crude, but they work well in practice. The order of convergence is (1 + /5)/2 1.6 for a single variable. Their convergence is slightly slower than a properly chosen finite difference Newton method, but they are usually more efficient in terms of total function evaluations to achieve a specified accuracy (see Dennis and Schnabel, 1983, Chapter 2). 

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

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

Procedures that compute a search direction using only first derivatives of/provide an attractive alternative to Newton s method. The most popular of these are the quasi-Newton methods that replace H(x ) in Equation (6.11) by a positive-definite approximation W ... 

Shanno, D. F. Conditioning of Quasi-Newton Methods for Function Minimization. Math Comput 24 647-657 (1970). 

Broyden, C. G. Quasi-Newton Methods and Their Application to Function Minimization Math Comput 21 368 (1967). 

Cite two circumstances in which the use of the simplex method of multivariate unconstrained optimization might be a better choice than a quasi-Newton method. 

For the quasi-Newton method discussed in Section 6.4, give the values of the elements of the approximate to the Hessian (inverse Hessian) matrix for the first two stages of search for the following problems ... 

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