Normal solutions of linear programs

Abstract

The solvability of a linear program is characterized in terms of the existence of a fixed projection on the feasible region, of all sufficiently large positive multiples of the gradient of the objective function. This projection turns out to be the normal solution obtained by projecting the origin on the optimal solution set. By seeking the solution with least 2-norm which minimizes the l-norm infeasibility measure of a system of linear inequalities or of the optimality conditions of a linear program, one is led to a simple minimization problem of a convex quadratic function on the nonnegative orthant which is guaranteed to be solvable by a successive overrelaxation (SOR) method. This normal solution is an exact solution if the original system is solvable, otherwise it is an error-minimizing solution. New computational results are given to indicate that SOR methods can solve very large sparse linear program that cannot be handled by an ordinary linear programming package.

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grant MCS-8200632.