Ascent, steepest

One strategy that has often been used is to proceed along the path of steepest ascent until a maximum is reached. Then another search is made. A path of steepest ascent is determined and followed until another maximum is reached. This is continued until the climber thinks he is in the vicinity of the global maximum. To aid in reaching the maximum, the technique of using three points to estimate a quadratic surface, as was done previously, may be used. 

One problem with this or any other method using gradients is that the best path obtained is dependent on the units used. If different units are used a different path will be indicated. To illustrate this, suppose it is desired to improve the yield (y) of a plug flow reactor when the feed rates and compositions are constant. At the usual operating conditions of 50 psia and 500°K a yield of 60 lb/hr is obtained. In what order should the pressure (P) and the temperature (T) be changed To reduce costs, it is desirable to minimize the number of experiments performed, hence the method of steepest ascent is to be used.When a test is performed at 50 psia and 510°K, the yield is found to be 60 lb/hr. When another experiment is run at 60 psia and 500°K, the yield is again 60 lb/hr. If the surface is linearized it can be expressed as ... 

The path of steepest ascent is the one for which the pressure increases one psia for each degree Kelvin the temperature changes. If the pressure units were not psia but atmospheres, then Equation 7 would become... 

Figure 14-2 Three different paths of steepest ascent which result from using the same data but different units.

Kelvin equivalent to 1 psia, 1 atm, 1 mm Hg, or some other pressure unit No definite answer can be given. Since this is always true, the method of steepest ascent can only tell whether a variable should be increased or decreased. It cannot tell how much one independent variable should be moved in relation to another one having different units. 

Figure 6.10 The topological map of an idealized mountain represented by the circular contours of constant height on a topological map. Two gradient paths or lines of steepest ascent (a) are shown, together with a path (b) that is not a line of steepest ascent but is an easier route up the mountain. The lines of steepest ascent—gradient paths—cross the contours at right angles.

For a homonuclear diatomic molecule such as Cl2 the interatomic surface is clearly a plane passing through the midpoint between the two nuclei—in other words, the point of minimum density. The plane cuts the surface of the electron density relief map in a line that follows the two valleys leading up to the saddle at the midpoint of the ridge between the two peaks of density at the nuclei. This is a line of steepest ascent in the density on the two-dimensional contour map for the Cl2 molecule (Fig. 9). 

The well-known Box-Wilson optimization method (Box and Wilson [1951] Box [1954, 1957] Box and Draper [1969]) is based on a linear model (Fig. 5.6). For a selected start hyperplane, in the given case an area A0(xi,x2), described by a polynomial of first order, with the starting point yb, the gradient grad[y0] is estimated. Then one moves to the next area in direction of the steepest ascent (the gradient) by a step width of h, in general... 

One important use of experimental designs is to achieve optimum operating conditions of industrial processes. For a discussion of this application, see Box and Wilson (1951). This paper is extraordinarily rich in response surface concepts. What is the steepest ascent technique discussed in this paper What models are assumed, and what experimental designs are used ... 

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. 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

Optimization Steepest ascent k factors Follow-up on 2-level design ... 

An adaptation of the Box method, however, seems to offer the advantage of improved efficiency while still being susceptible to automatic computation. Box s approach may be divided into two stages. The first, to which he has applied the name method of steepest ascents, is primarily for the purpose of approximately locating the optimum response. The second is a more intensive investigation in the local region of the optimum. This will permit a precise determination of the optimum and also indicate the behavior of the response in its neighborhood. 

The basis for the method of steepest ascents is easy to appreciate for a... 

Fig. 3. The method of steepest ascents applied to a problem of maximizing a response of two variables.

It is easy to imagine how the method of steepest ascents is generalized to multidimensional studies. It is, however, difficult to portray these cases graphically. They are the most important applications for the method and it is in such large studies that the technique is most advantageous. 

The mechanics of the method of steepest ascents are simple. The first step is to define the general area of interest and reduce all the controllable... 

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. 

Following application of the method of steepest ascents, it will usually be advisable to investigate the neighborhood of the optimum more carefully, fitting at least a second-degree polynomial to the response surface as described in Section VI B. 

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