Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution

Abstract

In order to express a polyhedron as the Minkowski sum of a polytope and a polyhedral cone, Motzkin (Beitrage zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) devised a homogenization technique that translates the polyhedron to a polyhedral cone in one higher dimension. Refining his technique, we present a conical representation of a set in the Euclidean space. Then, we use this representation to reach four main results: First, we establish a convex programming based framework for determining a maximal element—an element with the maximum number of positive components—of a non-negative convex set—a convex set in the non-negative Euclidean orthant. Second, we develop a linear programming problem for finding a relative interior point of a polyhedron. Third, we propose two procedures for identifying a strictly complementary solution in linear programming. Finally, we generalize Motzkin’s (Beitrage zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) representation theorem for a class of closed convex sets in the Euclidean space.