Polak-Ribiere

For quadratic functions this is identical to the Fletcher-Reeves formula but there is some evidence that the Polak-Ribiere may be somewhat superior to the Fletcher-Reeves procedure for non-quadratic functions. It is not reset to the steepest descent direction unless the energy has risen between cycles. 

---

IlyperChem supplies three types of optimi/ers or algorithms steepest descent, conjugate gradient (Fletcher-Reeves and Polak-Ribiere), and block diagonal (Newton-Raph son). 

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. 

Choose algorithm such as Steepest descent, Fletcher-Reeves (conjugate gradient), or Polak-Ribiere (conjugate gradient, default of FlyperChem), and choose options for termination condition such as RMS gradient (e.g., 0.1 kcal/mol A) or number of maximum cycles. 

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 methods1-175 178]. 

Three of the best known settings for (3 are titled the Fletcher-Reeves (FR), Polak-Ribiere (PR), and Hestenes-Stiefel (HS) formulas.66-71 77 78 They are given by the formulas... 

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. 

Points-On-a-Sphere (POS) force held, 39 Polak-Ribiere conjugate gradient optimization, 318 Polaiizabihty, 236, 248... 

The analysis, which was done after calculation, showed the absence of false fi equencies for all investigated molecules. It confirms that points corresponding to minimal potential energy have been found. The analogous calculations with the use of HyperChem 6.0 and semi empirical approximation PM3 have been carried out for the comparison, with the above-mentioned results. Optimizing of the molecular structure was carried with use of the Polak-Ribiere algorithm until the minimal potential energy was reached. The accuracy of calculation was not less then 10 kcal/A-mol. 

---