Duality in linear programming: From trichotomy to quadrichotomy

Abstract

In this paper, we present a new approach to the duality of linear programming. We extend the boundedness to the so called inclusiveness, and show that inclusiveness and feasibility are a pair of coexisting and mutually dual properties in linear programming: one of them is possessed by a primal problem if and only if the other is possessed by the dual problem. This duality relation is consistent with the symmetry between the primal and dual problems and leads to a duality result that is considered a completion of the classical strong duality theorem. From this result, complete solvability information of the primal (or dual) problem can be derived solely from dual (or primal) information. This is demonstrated by applying the new duality results to a recent linear programming method.