Increasing the efficiency of selected grid search procedures through the introduction of a best step mechanism

In this paper, several new developments are suggested in order to increase the efficiency of grid search procedures used within vertex enumeration solution algorithms based on both the classical cobweb technique (a two constraint approach) and a single constraint approach. The gains in efficiency (and robustness) are demonstrated by incorporating the developments into a grid search procedure (which is a part of a heuristic solution algorithm used to obtain solutions to a nonlinear bilevel multiperiod model of an aluminium smelter) and applying it to a test suite of problems. The classical cobweb technique is shown to have some serious problems when used in this application but with step size improvements suggested in the paper is shown to be very competitive. The developments are also incorporated into a single constraint based approach which does not suffer from the cobweb specific difficulties. The improvements considered include the use of a step mechanism initially with a step size calculated a priori and then finally with a tailored (best) one. The latter development when applied to the cobweb approach requires the approximation of the two constraints (and hence the intersection of their graphs) using second degree polynomials.