Direction of steepest

Steepest Descent is the simplest method of optimization. The direction of steepest descen t. g, is just th e n egative of the gradien t vector ... 

Ok) function is sought by repeatedly determining the direction of steepest descent (maximum change in for any change in the coefficients a,), and taking a step to establish a new vertex. A numerical example is found in Table 1.26. An example of how the simplex method is used in optimization work is given in Ref. 143. 

Initial search is in the direction of steepest descent given by the reduced gradient, z0 say. Subsequent search directions sk+, are generated by a conjugate direction formula (F6),... 

From one viewpoint the search direction of steepest descent can be interpreted as being orthogonal to a linear approximation (tangent to) of the objective function at point x examine Figure 6.9a. Now suppose we make a quadratic approximation of/(x) at x ... 

The key to investigating the topology of the electron density p is the gradient vector V p, which is perpendicular to a constant electron density snrface and points in the direction of steepest ascent. Then, a sequence of infinitesimal gradient vectors corresponds to a gradient path. Since gradient vectors are directed, gradient paths also have a direction They can go uphill or downhill. 

Once the transformations have been made, a solution space has been defined. It is only within this framework that the concept of steepest ascents takes meaning. The proper direction in which to proceed can be determined by n + 1 cases if n is the number of controllable variables. In each of the n cases one variable is changed slightly from its value in the base case, while all other variables are held constant. This permits approximating the n partial derivatives of response with respect to each variable. The direction of steepest ascent is given by the vector which is the gradient of the response, R ... 

Before proceeding further it is necessary to decide upon the size of the step to be made in the direction of steepest ascent. This step size, A, can be related to the several partial derivatives by a constant K ... 

Once the constant K has been determined from this equation, the incremental change in each individual variable for steps in the direction of steepest ascent is... 

Now a new case is calculated at the best previous point and a new direction of steepest ascent is determined. The process is repeated as many times as seems advisable. The entire procedure is susceptible to automatic treatment on the computer. The only point of uncertainty is the size of the steps to be taken. It may be necessary to revise this from time to time depending upon the progress of the study. As the optimum is approached, the steps should decrease in size. Even if it is decided that for a particular problem and a particular computer completely automatic calculation is impractical, at least some fairly large combination of operations can be programmed for one computer run. 

The steepest descent method193-961 is a simple gradient method (Fig. 3.6). At each point the first-derivative vector A is used to find the direction of steepest descent (Eq. 3.1 and 3.2), where A, is the step size. 

The pseudoinverse for the calculation of the shift vector has been computed traditionally as J+ = -(JlJ) Jl. Adding a certain number, the Marquardt parameter mp, to the diagonal elements of the square matrix JlJ prior to its inversion has two consequences (a) it shortens the shift vector 5k and (b) it turns its direction toward the direction of steepest descent. The larger the Marquardt parameter, the greater is the effect. 

When applied to a mountain, V g gives the direction of steepest descent for the most ardent and risk-averse skier ... 

This particular NLLSQ program used the Marquardt method of expanding a model in a truncated Taylor s type series and solves for improved estimates of parameters in an iterative manner. The very nature of this strategy is that it finds a particular direction to move in search of better parameter estimates which is not exactly like the normal truncated Taylor s series nor that of the direction of steepest descents". 

No sane cartographer, in drawing a contour map of a hill, would ever choose a scale in the north-BOuth direction different from that in the east west direction. The concept of length, well-defined in the science of geometry, forces him to make the scale independent of the direction. Consequently any cartographer is within his rights to draw small circles and to speak of a direction of steepest ascent. 

Figure 2 shows a yield contour for a particular choice of scales for and x2. The same contour is plotted in Fig. 3 for which the horizontal scale has been doubled. The contour tangent and gradient line at the same point a are given for each choice of horizontal scale. The two gradient lines obviously do not contain the same points. Thus the direction of steepest ascent depends entirely on the relative scales of the... 

If we move on the activation surface A, the energy will remain Qq. If we move in the direction from which the path defining the activated state comes, or if we move in the direction of this path, the energy will decrease, because the activation point is the highest one of this path. In all directions perpendicular to the direction of steepest descent, however, El will increase. Otherwise, the highest point of an adjoining... 

We can now draw a 3n — i dimensional surface F through the activated points, perpendicular always to the direction of steepest descent. A will have, in general, less than 3 — i dimensions and lie entirely in F [cf. Fig. i). For a reaction in a system to occur, it is necessary that the representative point shall cross this n — i dimensional surface. Most crossings will occur, of course, in the neighbourhood of the activation points, at any rate at such points the enei y of which is not many kT greater than the activation energy Qq. It is easy to calculate the number of systems of our assembly which cross F in one direction in unit time. It is simply equal to the density of points on the surface multiplied by the mean velocity perpendicular to the surface, and this integrated all over the surface ... 

Comparing (5.13) and (5.12) we deduce that l(m) really is the direction of steepest ascent,... 

This result comes from the simple fact that if we minimize a functional along some direction, described by a parametric line, the direction of steepest ascent must... 

By taking into account the direction of steepest descent, the asymptotic diabatic wave functions at x —> OC can be expressed as a sum of contributions from... 

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