Ascent path

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. 

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. 

U. Different directions of the steepest ascent path due to scaling of the variahles... 

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 ... 

Fig. 6.3. Response contour lines for the plane described by Eq. (6.3). The arrow indicates the steepest ascent path starting at the center of the design. The values in parentheses are the experimentally determined responses.

Points on the steepest ascent path for the models of Figs. 6.2 and 6.3... 

Exercise 6.5. Imagine that, in the C. elegans example, the researchers had preferred to take the concentration of glucose as the starting factor to determine the steepest ascent path, with an initial displacement of +25 gL (note that these are the units actually used in the laboratory). Calculate the coordinates of the third point along the new path, and use Eq. (6.5) to predict chitin yield imder these conditions. 

Fig. 6.4. Results obtained in the runs performed along the steepest ascent path of Fig. 6.3.

It is clear that C (X, i ) cannot be a single point catchment region, since there cannot exist a minimum of E(K) on the boundary B(A,i) of any catchment region C(X,i). Furthermore, C (X., i ) cannot contain points exclusively from the boundary B(X,i), since any infinitesimal neighborhood of K(X, i ) must cut into C(X,i), from where a steepest descent path leads away from K(X, i ). Hence, for the inverted surface -E(K), the corresponding path, when reversed, is a steepest descent path that leads toward K(X,, i ). Consequendy, there must exist some steepest ascent path of E(K) that leads from an interior point K of C(X,i) to K(A,, i ) on the boundary B(A,i). We conclude that this point K must lie within both catchment regions C(X,i) and C (, i ), hence their intersection is not empty. 

STEEPEST DESCENT PATHS FOR UNCONSTRAINED SYSTEMS [3,18,22] (for ascent paths, see Refs.[23-26])... 

In general, both methods do not yield the same results. The second method (ii) produces additional curves which are other (artificial) solutions of the GE equation which do not characterize valley floors. It should be noted that there are some simplifications of (ii) when following a given eigenvector of the Hessian by the so-called stream bed method, see [3-5]. SDP and least ascent path (VF-GE) have each certain advantages as well as limitations. 

A peculiarity of an ascent path is that branching and dissipation points (critical points, see Sects. 2.6 and 3.2) which characterize the regions of branching or dissipation of a valley, may make the path tracing more difficult by the occurrence of zeros of the Hessian matrix. On the other hand, this may promote the development of procedures which allow to locate branching points (cf.Ref.l9). 

In searching for new and still more complete RP definitions, the problem consists in finding an ascent path along the direction(s) characterizing the actual atomic movements connected with the reaction under investigation or a steepest descent path which overcomes the problems near the reactant minimum. The descent path is the more advantageous one, first of all because of the inclusion of the SP of interest by definition (a path in ascent may find any other SP). 

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