On linear systems containing strict inequalities

Abstract

This paper deals with systems of an arbitrary (possibly infinite) number of both weak and strict linear inequalities. We analize the existence of solutions for such kind of systems and show that the large class of convex sets which admit this kind of linear representations (i.e., the so-called evenly convex sets) enjoys most of the well-known properties of the subclass of the closed convex sets. We also show that it is possible to obtain geometrical information on these sets from a given linear representation. Finally, we discuss the theory and methods for those linear optimization problems which contain strict inequalities as constraints.