A Steepest Feasible Direction Extension of the Simplex Method

Abstract

We present a feasible direction approach to general linear programming, which can be embedded in the simplex method although it works with non-edge feasible directions. The feasible direction used is the steepest in the space of all variables, or an approximation thereof. Given a basic feasible solution, the problem of finding a (near-)steepest feasible direction is stated as a strictly convex quadratic program in the space of the non-basic variables and with only non-negativity restrictions. The direction found is converted into an auxiliary non-basic column, known as an external column. Our feasible direction approach allows several computational strategies. First, one may choose how frequently external columns are created. Secondly, one may choose how accurately the direction-finding quadratic problem is solved. Thirdly, near-steepest directions can be obtained from low-dimensional restrictions of the direction-finding quadratic program or by the use of approximate algorithms for this program.