Polak-Ribiere method

For a purely quadratic function the Polak-Ribiere method is identical to the Fletcher-Reeves algorithm as all gradients will be orthogonal. However, most functions of interest, including those used in molecular modelling, are at best only approximately quadratic. Polak and Riviere claimed that their method performed better than the original Fletcher-Reeves algorithm, at least for the functions that they examined. 

The computation involved in each cycle is more complex and time-consuming than for the steepest-descent method, but convergence is generally more rapid. Two commonly used examples are the Fletcher-Reeves and the Polak-Ribiere methods [232-235]. 

HyperChem provides two versions of the conjugate gradient method, Fletcher-Reeves and Bolak-Rihiere. Polak-Ribiere is more refined and is the default eh oiee in HyperChem,... 

There are several ways of choosing the /3 value. Some of the names associated with these methods are Fletcher-Reeves, Polak-Ribiere and Hestenes-Stiefel. Their definitions of /3 are... 

The Polak-Ribiere prescription is usually preferred in practice. Conjugate gradient methods have much better convergence characteristics than the steepest descent, but they are again only able to locate minima. They do require slightly more storage than the steepest descent, since the previous gradient also must be saved. 

Scales (1986) recommends the Polak Ribiere version because it has slightly better convergence properties. Scales also gives an algorithm which is used for both methods that differ only in the formula for the updating of the search vector. 

The Polak-Ribiere conjugate gradient method " was used in RSll to perform the non-linear mnlhvariate optimization of the objective function with the weighing factors, Wj = 1 and W2 = 10. 

An alternative formula for P, is the Polak-Ribiere formula see Leach, p. 225.) The idea of the conjugate-gradient method (which really should be called the conjugate-direction method) is to choose each new step in a direction that is conjugate to the directions used in the previous steps (where the word conjugate has a certain technical... 

HyperChem s optimizers (steepest descent, Fletcher-Reeves, and Polak-Ribiere conjugate-gradient methods, and the block diagonal Newton-Raphson) differ in their generality, convergence properties, and computational requirements. They are unconstrained optimization methods however, it is possible to restrain molecular mechanics and quantum mechanics calculations in HyperChem by adding extra restraining forces. 

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