Gradient method steepest descent

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

If there is no approximate Hessian available, then the unit matrix is frequently used, i.e., a step is made along the gradient. This is the steepest descent method. The unit matrix is arbitrary and has no invariance properties, and thus the... 

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

A con jugate gradicri I method differs from the steepest descent technique by using both the current gradient and the previous search direction to drive the rn in im i/ation. , A conjugate gradient method is a first order in in im i/er. 

The gradient at the minimum point obtained from the line search will be perpendicular to the previous direction. Thus, when the line search method is used to locate the minimum along the gradient then the next direction in the steepest descents algorithm will be orthogonal to the previous direction (i.e. gk Sk-i = 0)-... 

Tafe/e 5.1 A comparison of the steepest descents and conjugate gradients methods for an initial refinement and a stringent minimisation. 

This study shows that the steepest descent method can actually be superior to conjugate gradients when the starting structure is some way from the minimum. However, conjugate gradients is much better once the initial strain has been removed. 

The steepest descent method is a first order minimizer. It uses the first derivative of the potential energy with respect to the Cartesian coordinates. The method moves down the steepest slope of the interatomic forces on the potential energy surface. The descent is accomplished by adding an increment to the coordinates in the direction of the negative gradient of the potential energy, or the force. 

Example Compare the steps of a conjugate gradient minimization with the steepest descent method. Amolecular 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 minimum(E) because the optimizer remembers the penultimate step. 

Steepest Descent is the simplest method of optimization. The direction of steepest descent, g, is just the negative of the gradient vector ... 

Techniques used to find global and local energy minima include sequential simplex, steepest descents, conjugate gradient and variants (BFGS), and the Newton and modified Newton methods (Newton-Raphson). 

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 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. 

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