Linear programming by an effective method using triangular matrices

Abstract

A modification of the projection method for linear programming is presented. This modification determines the step direction by solving two triangular systems of linear equations. The triangular matrix is updated in each step by deleting a row and adding a new one whose elements were already computed for the step-size determination. Thus there is no real computational effort in the matrix-updating. The size of the triangular systems depends on how many of the active constraints have become active after the constraint that’s going to become inactive. In the worst case, i.e. if the oldest active constraint becomes inactive, the computational effort in solving the triangular systems corresponds to that of the matrix-updating in the projection method, whereas in all other cases the effort is reduced. This reduction can be very high. Cycling of the method is excluded by a very simple rule.