Newton-like method

Algorithmic Details for NLP Methods All the above NLP methods incorporate concepts from the Newton-Raphson method for equation solving. Essential features of these methods are that they rovide (1) accurate derivative information to solve for the KKT con-itions, (2) stabilization strategies to promote convergence of the Newton-like method from poor starting points, and (3) regularization of the Jacobian matrix in Newton s method (the so-called KKT matrix) if it becomes singular or ill-conditioned. 

Lucia, A., and Xu, J., Chemical process optimization using Newton-like methods, Comp, and Chem. Engr. 14(2), 119(1990). 

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable... 

In summary, therefore, there is too little work with Newton-like methods to make any assertion about their utility in quantum chemistry, but there is enough work with variable-metric methods to make it possible to assert with some confidence that they are worth very serious consideration by any worker wishing to optimize orbital exponents or nuclear positions in a wavefunction. 

Non-linear programming is a fast growing subject and much research is being done and many new algorithms appear every year. It seems to the Reporters that the current area of major interest in the field is the area of variable-metric methods, particularly those not needing accurate linear searches. Unfortunately, from a quantum chemical point of view, such methods are liable to be of use only in exponent and nuclear position optimization and in this context, as we have seen, Newton-like methods are also worth serious consideration. 

Ponder JW, FM Richards (1987) An Efficient Newton-Like Method for Molecular Mechanics Energy Minimization of Large Molecules. J. Comput. Chem. 8 (7) 1016-1024. (See http //dasher.wustl.edu/tinker/)... 

Even if a quasi-Newton method is used, a good approximation of the Jacobian is known. Consequently, this criterion is reliable enough. Nonetheless, it should be stressed that this test is correct only when the difference between two consecutive iterations, d = x +i — x , comes from a Newton-like method and the Jacobian is nonsingular. 

We start with a short discussion on the both ways that allow a determination of stationary points. Firstly, stationary points can be determined by solving Eg.(4). But, since the solution set of this equation contains the minimizers, the maximizers as well as the saddle points of E, an additional examination is necessary to determine the type of the solution when an unspecific (Newton-like) method is used,... 

This subsection is engaged in a unified approach to the Newton-like methods. 

As a consequence of proposition 3 we obtain In the vicinity of a minimizer of E the Newton vectors and the steepest descent vectors always point to the minimizer. In the vicinity of a saddle point the Newton vectors always point to the saddle point whereas a steepest descent vector point to the saddle point only if E is convex along that vector. This observation forms the basis for a modified Newton-like method which looks for stationary points of prescribed type (see Sect.2.4.3). 

According to the kind of modification we distinguish four types of Newton-like methods ... 

In practice the Newton-like methods do not occur in "pure form. Mostly effective procedures involve elements of several basic methods. 

Due to the nonlinearity of the discretized convective terms, resp., of the reinitialization step, iterative defect correction or Newton-like methods, resp., corrections via redistancing, must be invoked in steps (1) and (4). However, due to the assumed relatively small time steps, such nonlinear iteration methods are not critical for the complete flow simulation. 

This is now a linear differential equation in terms of the lower case correction function. Assuming this linear equation can now be solved, the correction function (lower case u) ean then be added to the approximation function (upper ease U ) and an improved solution obtained. The procedure can then be repeated as many times as necessary to achieve a desired degree of aeeuracy in the solution. It ean be seen that if a valid solution is obtained, the F funetion in Eq. (11.40) ap-proaehes zero so the correction term will approach zero. As with other Newton like methods, the solution is expected to converge rapidly as the exaet solution is approaehed. This technique is frequently referred to as quasilinerization and it has been shown that quadratic convergence occurs if the procedure eonverges. 

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