Linear programming under uncertainty: A basic property of the optimal solution

Summary We show that if the two-stage linear programming problem under uncertainty has an optimal solution, then it has an optimal solution in which the column vectors corresponding to the positive first stage decision variables are linearly independent. This leads to the result that if an optimal solution exists, then there exists an optimal solution in which not more than m + ¯m of the first stage decision variables are positive. These results extend to the multi-stage case.This research was partially supported by the Office of Naval Research under Contract Nonr-222(83), the National Science Foundation under Contract GP-4593, the Army Research Office under Contract DA-31-124-ARO-D-331 and the University of California. Reproduction in whole or part is permitted for any purpose of the United States Government.