The method of steepest ascent

The effects of the two factors can be separated as described in Section 7.7. Rewriting the table above, in the notation of that section, gives  

This method gives satisfactory progress towards the maximum provided that, over the region of the factorial design, the contours are approximately straight. This 

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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 ... 

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. 

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... 

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. 

Fig. 4. Possible erroneous optimum indication in using the method of steepest ascents.

Figure 6.19. Sequential application of complete factorial designs in the same situation as in Figure 6.18a. An alternative approach is the method of steepest ascent indicated by the dotted line.

A process having properties dependent on four factors has been tested. A full factorial experiment and optimization by the method of steepest ascent have brought about the experiment in factor space where only two factors are significant and where an inadequate linear model has been obtained. To analyze the given factor space in detail, a central composite rotatable design has been set up, as shown in Table 2.152. 

The property of the method of steepest ascent lies in the fact that movement along the gradient of a function must be preceded by a local description of the response surface by means of full or fractional factorial experiments [49]. It has been demonstrated that by processing FUFE or FRFE experimental outcomes we may obtain a mathematical model of a research subject in the form of a linear regression ... 

In practice, we have often met the case when the method of steepest ascent is deformed in a search for optimum with only one factor (one-dimensional optimiza-... 

Special attention is paid to replication of trials when applying the method of steepest ascent. Trials with best response values are in principle replicated only, although it is not pointless to replicate all trials. 

An Example of practical application of the method of steepest ascent has been demonstrated in a chemical-technological process. System response depended on xrratio of solvent to basic material, g/1 x2-temperature of reaction mixture, °C and x3-reaction time, min. The system response has been the yield of a pharmaceutical product (carbo-methoxysulphanyl guanidine) in per cent. Based on theoretical knowledge the yield may reach 95%, but in practice only a half of this has been reached. 

An analysis of outcomes of movement along the gradient, or of abstract trials, indicates an efficient application of the method of steepest ascent since the obtained yield is 72.5%. It is pointless to continue movement to optimum because smaller yields are obtained. The obtained optimum is probably local since even that value of yield is far away from theoretical. The difference between the achieved and theoreti-... 

The method of steepest ascent has proved efficient as the yield value is 82.4%. The linear model is symmetrical with respect to regression coefficients so that no change in the factor-variation interval is necessary. A negative value of regression coefficient bj is interesting from the technological point of view, for it had an effect on the fall of the xj factor value, which is desirable. 

The method of steepest ascent has shown that the maximal obtained value of tensile strength of adhesion is 36.1 kp/cm2. The best value in the basic 27"3 experiment was 45.6 kp/cm2 (Table 2.118). This means that optimum is either in the experimental region or in its vicinity. 

The Method of steepest ascent, Table 2.190, was applied to outcomes of a fractional factorial experiment 2s"4, Table 2.95. The experiment included eight factors xrquan-tity of binder, % x2-quantity of linen fabrics, g/m2 x3-pressure of pressing, kp/cm2 X4-temperature, °C xs-time of thermic processing, min x6-time of pressing, min xytype of binder, [1] and xg-quantity of dibutylphtalate, %. The system response was the relative elongation at strain yu %. 

Regression coefficients b2, b5 and b7 are statistically insignificant so that associated factors when applying the method of steepest ascent are fixed at corresponding levels. Other significant regression coefficients are symmetrical, which has been proved by successful application of the method of steepest ascent. Due to the fact that the optimum is in the vicinity of the experimental region, it is possible to switch to a second-order model. 

After the end of the application of the method of steepest ascent it is necessary to make conclusions. This depends on the success of the applied method. Based on this, we can differentiate these situations ... 

Assume this situation the basic design half-replica, linear model, is inadequate, the method of steepest ascent proved to be inefficient, the optimum area is close by. The system response is the product yield. Maximal possible yield is 100%. The best yield in realizing half-replica is 80%. Trial error is 1%. 

If the optimum area is far away and the linear model adequate, there exist good reasons for the method of steepest ascent to be successful. A possible explanation for a failure in applying the gradient method may lie in the form of the response surface with one extreme. The response surface may in reality have the form shown Fig. 2.47. 

In such a case, basic design should be moved to another part of factor space (II), and then apply the method of steepest ascent. 

The Method of steepest ascent has been applied again and the obtained outcomes are in Table 2.198. 

The method of steepest ascent has proved to be successful since we obtained a yield of 88.0% in trial No. 5. The optimum area is dose by and we may switch to a second-order design. CCRD 22+2 2+5 has been realized, Table 2.199. 

An analysis of outcomes after application of the method of steepest ascent clearly shows that trial No. 19 gives the best value for D3 and D2. Trial No. 24 gives the best value for D2 but D j has a lower value. The conditions of trial No. 19 have been chosen for the center of new FUFE 23 design for three most important factors X3 X3 and X4. The design matrix with outcomes of the experiment are given in Table... 

The linear regression model is inadequate with 95% confidence. Since the linear model is neither symmetrical nor adequate and since the application of the method of steepest ascent would lead to a one-factor optimization (b2 is by far the greatest), a new FRFE 24 1 has been designed with doubled variation intervals for X3 X3 and X4. 

The obtained regression coefficients (Table 2.204) are relatively symmetrical so that we applied the method of steepest ascent the results are in Table 2.204. 

Using data from Problem 2.19, apply the method of steepest ascent. 

Set up CCRD after application of the method of steepest ascent in trial with the lowest density. 

Typical situations have been studied by the analysis of application of the method of steepest ascent. Drawing conclusions and making decisions after application of the method depends on the success of its application, on the position of optimum and on the lack of fit of the linear model. 

Movement to optimum is realized step by step in such a way that the vertex with the most inconvenient response value is successively rejected and a new point or vertex, which is the physical mirror image of the rejected vertex, is constructed. In the next step, the experiment is done in the new vertex, and then again the vertex with the most inconvenient response value is rejected and the procedure repeated. Movement to optimum is here realized after each step and not after a series of trials as in the method of steepest ascent. Simplex movement to optimum is geometrically in a zig-zag line, while the center of those simplexes moves along a line close to the gradient. The geometrical interpretation is given in Fig. 2.49. 

In a process of isomerization of a sulfanilamid compound, the methods of both the steepest ascent and simplex optimization have been analyzed. Trials were performed in a laboratory plant. Table 2.214 shows FUFE 22 with application of the method of steepest ascent. Maximal yield by this method was 80%. Table 2.215 shows the application of simplex method (k=2) to the same process. The position of initial simplex corresponds completely to the position of trials of factor design Fig. 2.54. 

Nine trials were done in simplex optimization and a top value of the yield was obtained in trial No. 5 or vertex C. The maximal yield by the method of steepest ascent was obtained in trial No. 8, which coincides with simplex optimization. It can be concluded that to reach the optimum by the method of steepest ascent, six trials were realized, while by the simplex method, five trials were realized. We should, however, remember that FUFE 22 has to be replicated once, so that the method of steepest ascent, in this case, requires 12 trials, which is considerably more than by the simplex method. 

Let us analyze the previous case by taking into account the third factor X,. The outcomes of FUFE 23 and the results of application of method of steepest ascent are given in Table 2.216. Thirteen trials were necessary to reach the maximal yield of 85.2%. The outcomes of the simplex method are in Table 2.217. Maximal yield after 14 trials is 85.0%. Approximately the same number of trials has been necessary by both methods to reach the optimum. It should be stressed once again that FUFE requires replications, so that to reach optimum by the method of steepest ascent, we need at least twice as many trials. Evidently, a half-replica instead of FUFE in the basic experiment may reduce the number of trials. However, there is a possibility of wrong direction of the movement to optimum due to the possible effects of interactions. 

Regression coefficient b33 is not great when compared to the linear one, so that movement to optimum by the method of steepest ascent can be recommended, as shown in Table 2.236. 

The obtained linear model proved to be inadequate, which means that the response surface is curved. The method of steepest ascent is applied to linear members of the regression mode], Table 2.242. 

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