Conjugate Descent method, optimal

Example Com pare the steps of a conjugate gradien t min im i/ation with the steepest descent method.. A molecular system can reach a potential minimum after the second step if the first step proceeds from, A to B. If the first step is too large, placing the system at D, the second step still places the system near the tninimum(K ) because the optim i/,er remembers the penultimate step. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

The basic difficulty with the steepest descent method is that it is too sensitive to the scaling of/(x), so that convergence is very slow and what amounts to oscillation in the x space can easily occur. For these reasons steepest descent or ascent is not a very effective optimization technique. Fortunately, conjugate gradient methods are much faster and more accurate. 

On the other hand, conjugate gradient methods. are more effective in locating the minimum energy structure. In this approach previous optimization information is utilized. The second and all subsequent descent directions are linear combinations of the previous direction and the current negative gradient of the potential... 

Steepest descents can in fact be beaten by the Conjugate Gradient method (Stich et al., 1989 Teter et al., 1989). Suppose the function to be optimized... 

An alternative when the size of the molecule prevents use of the quasi-Newton or Newton-Raphson methods is to use an optimization method that uses only the gradient and not the Hessian. Two such methods are the steepest-descent method and the conjugate-gradient method. 

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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