Steepest ascent, optimization method

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... 

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... 

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. 

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. 

In optimization of the process of obtaining novocaine, FRFE 24"1 with generating ratio X XjXzXj was chosen for the basic design of experiments. The system factors are x,-time of reaction, min x2-temperature, °C x3-surplus of sodium salt of paraa-minobenzoic acid, % and X4-concentration of sodium salt of paraaminobenzoic acid, %. The system response is the yield of chemical reaction %. Outcomes of basic experiment with application of method of steepest ascent are shown in Table 2.191. 

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. 

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. 

Experimental Optimization of Research Subject 427 Table 2.216 FUFE 23 and method of steepest ascent... 

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive. 

Liu, C., I. Chu, and S. Hwang. 2001. Factorial designs combined with the steepest ascent method to optimize serum-free media for CHO cells. Enzyme Microb Technol 28 314-321. 

With new synthetic methods, mechanistic details are still obscured. It is not likely that such details will be revealed until the preparative utility of the procedure has been demonstrated. This means that an optimization of the experimental conditions must generally precede a mechanistic understanding. Hence, the optimum conditions must be inferred from experimental observations. The common method of adjusting one-variable-at-a-time, is a poor strategy, especially in optimization studies (see below). It is necessary to use multivariate strategies also for determining the optimum experimental conditions. There are many useful, and very simple strategies for this sequential simplex search, the method of steepest ascent, response surface methods. These will be discussed in Chapters 9 - 12. 

The method of steepest ascent determines the direction from an initial experimental domain which has the steepest slope upwards along the response surface, and hence point towards the optimum conditions. A series of experiments can then be run along this steepest ascent vector. This will lead to rapid improvements. The method was introduced by Box and Wilson[l] and was the first method for systematic multivariate optimization experiments in chemistry. 

The method of steepest ascent and the simplex search can handle only one criterion, while the resportse surface methods allow simultaneous mapping of several responses. Response surface modelling can therefore be used to optimize several responses simultaneously. The problem of multiple responses is elegantly handled by PLS modelling. This technique is discussed in Chapter 17. 

Using a direct search technique on the performance index and the steepest ascent method, Seinfeld and Kumar (1968) reported computational results on non-linear distributed systems. Computational results were also reported by Paynter et al. (1969). Both the gradient and the accelerated gradient methods were used and reported (Beveridge and Schechter, 1970 Wilde, 1964). All the reported computational results were carried out through discretization. However, the property of hyperbolic systems makes them solvable without discretization. This property was first used by Chang and Bankoff (1969). The method of characteristics (Lapidus, I962a,b) was used to synthesize the optimal control laws of the hyperbolic systems. 

Most of the studies cited are based on the linearity or the availability of analytical solutions for the systems considered. Many engineering problems, mostly in the chemical engineering field, are non-linear and results must be obtained through computations. Several numerical studies have been reported in the literature on distributed systems. One of the early computational results on distributed optimization was given by Denn ei al. (1966). The solution of the linearized variational equations by Green s function that leads to both the necessary conditions and computational scheme based on the method of steepest ascent, was obtained. The computational difficulties due to the discretization scheme of the catalyst decay problem was overcome by Jackson (1967). Computational results based on steepest ascent for the optimal control for a non-linear distributed systems were also reported by Denn (1966). 

The centre of interest introduced in paragraph B corresponds either to conditions already tested in practice (usual manufacturing conditions) or to the result of an optimization using one of the direct methods described in chapter 6 (simplex, steepest ascent, optimum path). 

The implicit approach of chapter 5 was to optimize the process or formulation by examining the response surface directly. But other methods are both useful and necessary when there are many factors (desirability) or when the optimum is outside the experimental region (steepest ascent and optimum path). Also there are direct optimization methods available (sequential simplex) which do not involve mapping the response surface at all. 

If the goal of the experimentation has been to optimize something, the next step after analysing the results of a 2 or a fractional 2 design is to try to make improvement using knowledge provided by the analysis. The most common technique is the method of steepest ascent, also called the gradient (path) method. 

To test the model performance, experimental data gathered with Tank Oil were used to fit these 3 parameters. An optimization program was written to obtain the three parameters. For the initial guesses of the parameters and 5 literature values were used. Method of steepest ascent was used to minimize the following objective function, F ... 

There are yet further sophistications such as the supermodifled simplex, which allows mathematical modeling of the shape of the response surface to provide guidelines as to the choice of the next simplex. Simplex optimization is only one of several computational approaches to optimization, including evolutionary optimization, and steepest ascent methods, however, it is the most commonly used sequential method in analytical chemistry, with diverse applications ranging from autoshimming of instruments to chromatographic optimizations, and can easily be automated. 

It is evident that factorial design and the method of steepest ascent will be very complicated when several factors are involved. The next section describes a method of optimization which is conceptually much simpler. 

It can be seen that in contrast to factorial designs used in the method of steepest ascent, the number of experiments required in the simplex method does not increase rapidly with the number of factors. For this reason all the factors which might reasonably be thought to have a bearing on the response should be included in the optimization. 

In seeking the optimal point, it is common to use the method of steepest ascent or the gradient method. In this approach, based on the resulting relationship, the new process operating point wDl be determined based on the direction with the steepest gradient. This approach ensures that the optimal point can be reached the fastest. 

The basic difficulty with the steepest descent method is that it is too sensitive to the scaling of/(x), so that convergence is very slow and what amounts to oscillation in the x space can easily occur. For these reasons steepest descent or ascent is not a very effective optimization technique. Fortunately, conjugate gradient methods are much faster and more accurate. 

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