Steepest ascent method

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... 

As said before, linear models are used to reach (move towards) optimum, so that the significance of regression coefficients is an assumption for successful application of the steepest ascent method. Linear models, therefore, include as many factors as possible, and full factorial experiments are even replicated with increased factor variation intervals. 

There is only one solution then to examine reasons for inadequacy of the linear model according to block diagram Fig. 2.39. We can name several causes of unsuccessful application of the steepest ascent method ... 

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. 

The Steepest Ascent Method and Optimum Path Methods... 

Screening and factor studies will sometimes indicate whether, and if so, where we should search for an optimum within the domain being studied. However, if the optimum (we are considering a single key variable here) lies outside the present experiment, then the steepest ascent method comes into its own. The direction... 

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. 

When the optimum is outside the domain, we need to arrive at it rapidly. Changing one factor at a time will not work well, especially if there are interactions. One possibility is to use the sequential simplex, but it is here also that the steepest ascent method comes into its own. The procedure is simple. Assuming that the response (such as the yield) is to be maximized, we determine this response as a function of the coded variables x, x, . .. and find the position and direction of maximum rate of increase (steepest ascent) in terms of these coded variables. 

If the design was for a second-order model and examination of the contour plots or canonical analysis (see below) showed that the optimum probably lay well outside the experimental domain, then the direction for exploration would no longer be a straight line, as for the steepest ascent method. In fact, the "direction of steepest ascent" changes continually and lies on a curve called the optimum path. The calculations for determining it are complex, but with a suitable computer program the principle and graphical interpretation become easy. 

The starting simplex is usually regular, but does not have to be so. It is quite possible to select points from a factorial or fractional factorial design, with reduction of the number of factors (taking only those that are active) and using either the basic or extended sequential simplex methods to move from the region of the factorial design to a more favourable point. However under such circumstances the steepest ascent method could equally well be used, and this should normally be preferred. 

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. 

The gradient method, also called steepest descent or steepest ascent method, depending on whether one searches for a minimum or a maximum, is based on the following observation if it is possible to calculate the partial derivatives of the objective function S with respect to the parameters, or discrete approximations thereof, then for each parameter vector p, it can be calculated along which direction 5(b) changes fastest. This direction is given by the reverse of the gradient of 5 in b ... 

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