Fletcher-Reeves

Areturn to step 2 is made with the new conjugate direction. 

The conjugate direction is reset to the steepest descent direction every 3N search direction s or cycles, or if the energy rises between cycles. 

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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,... 

Several variants of the conjugate gradients method have been proposed. The formulatior given in Equation (5.7) is the original Fletcher-Reeves algorithm. Polak and Ribiere proposed an alternative form for the scalar constant 7) ... 

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. 

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... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

EXAMPLE 6.2 APPLICATION OF THE FLETCHER-REEVES CONJUGATE GRADIENT ALGORITHM... 

Results for Example 6.2 using the Fletcher-Reeves method... 

Search trajectory for the Fletcher-Reeves algorithm (the numbers designate the iteration). 

For problems jvith hundreds or thousands of variables, storing and manipulating the matrices H or V2/(x ) requires much time and computer memory, making conjugate gradient methods more attractive. These compute sk using formulas involving no matrices. The Fletcher-Reeves method uses... 

The one-step BFGS formula is usually more efficient than the Fletcher-Reeves method. It uses somewhat more complex formulas ... 

Use the Fletcher-Reeves search to find the minimum of the objective function... 

Find the minimum of the following objective function by (a) Newton s method or (b) Fletcher-Reeves conjugate gradient... 

The conjugate gradient method is one of the oldest in the Fletcher-Reeves approach, the search direction is given by... 

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. 

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