Method of steepest ascent

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. 

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... 

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.

Given the mere handful of reports in the published literature (6,38,39,52), there are many avenues open in the development of systematic approaches to optimization in SFC. In addition to the opportunities mentioned in the sections on the simplex method and window diagram approach, others include the exploration of other sequential or simultaneous optimization strategies such as optiplex, simulated annealing, method of steepest ascent, etc. that are potentially useful in SFC. 

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.

In Example 2.26, we have obtained the linear regression model for dynamic viscosity y, P, as a function of mixing speed X3, min"1 and mixing time X2, min of composite rocket propellant. To determine the conditions of minimal viscosity, a method of steepest ascent has been applied. This method has defined the local optimum region that has to be described by a second-order model. Conditions of the factor variations are shown in Table 2.146. 

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. 

A case of application of fractional replica 27-3 of a full factorial experiment on studying adhesion of thermoplastic polymer and fiber has been analyzed earlier in Example 2.33. Tensile strength of adhesion has been measured as the system response. The experiment included seven factors, with the nature of fiber being a qualitative-categorical factor. The regression coefficient values and method of steepest ascent are shown Table 2.188. 

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 ... 

Figure 2.45 Procedure after application of method of steepest ascent... 

If in realization of basic design of experiments high enough response values have been obtained, and if in application of method of steepest ascent no significant increase of this response occurred, one of these two solutions is accepted ... 

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. 

---