Steepest direction search algorithm

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. 

The stepping trajectory-based algorithms, such as simplex [30] and steepest descent [31] methods, are designed as more directed searches than the other types... 

The gradient at the minimum point obtained from the line search will be perpendicular to the previous direction. Thus, when the line search method is used to locate the minimum along the gradient then the next direction in the steepest descents algorithm will be orthogonal to the previous direction (i.e. gk Sk-i = 0)-... 

Steepest descents The steepest descents method is a minimization algorithm in which the line search direction is taken as the gradient of the function to be minimized. The steepest descents method is very robust in situations where configurations are far from the minimum but converge slowly near the minimum (where the gradient approaches zero). 

Fig. 3.6. Steepest descent algorithm (thin line) The derivative vector from the initial point Pq (Xq./q) defines the line search direction. The derivative vector does not point directly toward the minimum (O). The negative gradient of the...

The steepest descent algorithm is sure-fire. If the line minimization is carried out sufficiently accurately, it will always lower the function value, and is therefore guaranteed to approach a minimum. It has, however, two main problems. Two subsequent line searches are necessarily perpendicular to each other if there was a gradient component along the previous search direction, the energy could be further lowered in this direction. The steepest descent algorithm therefore has a tendency for each line search to partly spoil the function lowering obtained by the previous search. The steepest... 

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