Nondegeneracy of Polyhedra and Linear Programs

Abstract

This paper deals with nondegeneracy of polyhedra and linear programming (LP) problems. We allow for the possibility that the polyhedra and the feasible polyhedra of the LP problems under consideration be non-pointed. (A polyhedron is pointed if it has a vertex.) With respect to a given polyhedron, we consider two notions of nondegeneracy and then provide several equivalent characterizations for each of them. With respect to LP problems, we study the notion of constant cost nondegeneracy first introduced by Tsuchiya [25] under a different name, namely dual nondegeneracy. (We do not follow this terminology since the term dual nondegeneracy is already used to refer to a related but different type of nondegeneracy.) We show two main results about constant cost nondegeneracy of an LP problem. The first one shows that constant cost nondegeneracy of an LP problem is equivalent to the condition that the union of all minimal faces of the feasible polyhedron be equal to the set of feasible points satisfying a certain generalized strict complementarity condition. When the feasible polyhedron of an LP is nondegenerate, the second result shows that constant cost nondegeneracy is equivalent to the condition that the set of feasible points satisfying the generalized condition be equal to the set of feasible points satisfying the same complementarity condition strictly. For the purpose of giving a preview of the paper, the above results specialized to the context of polyhedra and LP problems in standard form are described in the introduction.