Path of steepest ascent

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. 

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.

Sometimes it is not necessary to determine a response surface model tor locate the optimum conditions. Hill-climbing by direct search methods, e.g. search along the path of steepest ascent [8] or sequential simplex search [9], will lead to a point on the response surface near the optimum. The computations involved in these methods are rather trivial and do not require a computer and will for this reason not be discussed further in this chapter. Readers who require details of these direct search methods should consult Refs. [1,8,9]. 

A fractional factorial design was employed in the preparation of effervescent aspirin tablets. The optimum conditions for preparing the tablets were determined following the path of steepest ascent. ... 

A series of experiments along the path of steepest ascent is obtained by... 

However, it is not to be taken for granted that the experiments in Table 10.3 really are located on the path of steepest ascent if the response surface model should be expressed in the natural variables. The slopes in different directions of the response surface are not invariant when the variables are transformed. When the coded variables, Xj, are translated back to the natural variables, u, the step of variation of the natural variables, Sj, will intervene and this may change the direction of the steepest ascent path. The direction is invariant to a change of variables, only if the steps of variation are equal for all variables. If there are different units of measurement, e.g. " C, h, equivalents of reagents, etc., they are likely to be different. 

It is seen in Fig. 10.5, that the path of steepest ascent will be in a direction which is perpendicular to the isoresponse lines. An equation for a line in the direction to the isoresponse lines will thus be... 

For this model, the iso-response surfaces are parallel hyperplanes in 4 dimensional factor space. In general, for k variables, the iso-response surfaces are - 1 dimensional hyperplanes. The path of steepest ascent is a straight line orthogonal to these planes, usually starting from the centre of the domain. The step-size for each variable is proportional to the estimate of the corresponding coefficient in the model. 

X3 may be neglected the frequency is thus held constant at 2000 Hz. It is X which has the greatest effect. As we have seen, step-sizes of the coded variables for the steepest ascent path are proportional to the coefficients bi of the fitted equation. The path of steepest ascent from the centre is ... 

The arrow in Fig. 6.3 indicates the path of steepest ascent starting at the center point. The direction of this path can be determined algebraically from the model coefficients. For maximum inclination, we must move in steps taken with a 62/ 1 ratio along the X2 and Xi axes. 

From Eq. (6.3) we have 62/ 1 = 4.25/(—5.25) =—0.81, which means that for each unit distance receded on the Xi axis we must advance 0.81 units along X2. The coordinates of several points along the path of steepest ascent are presented in Table 6.3, in both coded values and in the concentration and stirring speed units actually used in the laboratory. 

For the general case of a response surface determined by p factors, the steps along the path of steepest ascent are proportional to the magnitudes... 

Figure 7.8 Contour diagrams the arrow in each diagram indicates the path of steepest ascent. In [a] it goes close to the maximum but in [b] it does not.

ABSTRACT The strain with high endoinulinase productivity was screened, named G-60 which were collected from the root soil of Jerusalem artichoke in Shihezi city of Xinjiang province of China. A Plackett-Burman design was used to evaluate the influence of eight factors. The results showed that inulin added amount, content of the peptone and the Liquid medium volume played an important role in influencing the endoinulinase activity. Then the path of steepest ascent and the Central Composite Design were suitable for further optimization, the optimal concentration of the variables were determined as Inulin 6.65%, Peptone 2.91%, and liquid medium volume 50 mL under this condition, the endoinulinase activity was increased to 23.57 U/mL, which was improved by 1.55 times than before optimization. 

Table 5. The path of steepest ascent experiment design and response ...

Optimization of medium composition with response surface methodology Based on the previous Plackett-Burman screening test and the path of steepest ascent experiment design, response surface methodology was applied to determine the optimal conditions of these three significant factors, including initial inulin content, peptone, and liquid medium volume. 

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